Law of exponents worksheets | TPT - Free Printable

To solve this puzzle, we need to simplify each mathematical expression in the grid to find a single number. Then, we match that number to a letter using the provided key (A=0, B=1, C=2, ..., Z=25). Finally, we read the letters in order to form a word or phrase.

Here is the step-by-step simplification for each cell:

Row 1:
1. C: $3^8 \cdot 3^2$
* Rule: Add exponents when multiplying same bases ($a^m \cdot a^n = a^{m+n}$).
* $3^{8+2} = 3^{10}$. Wait, looking at the pattern of other answers, let's re-evaluate. Usually these puzzles result in small integers. Let's look closer. Ah, it might be division? No, it's a dot. Let's look at the next one.
* Let's check the value. $3^{10} = 59049$. This is way too big for the A-Z scale (0-25). Let me re-read the first term. Is it $3^8 / 3^2$? The image shows a dot. Let's look at cell W: $(3 \cdot 8)$. That equals 24. 24 corresponds to Y.
* Let's look at cell C again. Maybe it's not $3^8$. It looks like $3^8 \cdot 3^2$. Let's hold on this. Let's solve the easier ones first to see if a pattern emerges or if I'm misinterpreting the notation.
* Actually, let's look at the third cell in Row 1: $10^4$. That's 10,000. Also too big.
* Let's look at the fourth cell: $(\frac{2^9}{2})^4$. Inside parenthesis: $2^9 / 2^1 = 2^8$. Then $(2^8)^4 = 2^{32}$. Huge.
* Correction: I must be misreading the operations or the goal. Let's look at the bottom row. $6 \cdot 6^4 = 6^5 = 7776$.
* Let's look at cell A: $\binom{4}{5}^5$? No, that's not standard. It looks like $(\frac{4}{9})^5$? No.
* Let's look really closely at the image again.
* Cell A: $\left(\frac{4}{9}\right)$? No, it looks like $\binom{4}{5}$ is impossible. It looks like $(\frac{4}{?})$.
* Let's try a different approach. Look at Cell P: $\frac{7^8}{7^7}$. This simplifies to $7^{8-7} = 7^1 = 7$. Letter H.
* Cell B: $\frac{4^8}{5} \cdot 3$? No, it looks like $\frac{4^8}{5^3}$? Or maybe $\frac{4^3}{5}$? Let's assume the exponents are small.
* Let's look at Cell U: $2^7 \cdot 2^3 = 2^{10} = 1024$. Still huge.

Alternative Interpretation: Are these logarithms? No "log" written.
Are these modulo arithmetic? Unlikely for this level.

Let's look at the structure again.
Maybe the expressions simplify to numbers between 0 and 25.

Let's re-examine Cell W: $(3 \cdot 8) = 24$.
Key: 24 = Y.

Let's re-examine Cell F: $(4^1)^3 = 4^3 = 64$. Too big.
Wait, is it $(4/1)^3$? No.
Is it $4^{1/3}$? No.

Let's look at Cell T: $\frac{2^9}{2^4}$? The image shows $\left(\frac{2^9}{2}\right)^4$? No, the 4 is outside.
What if the top right is $\frac{2^9}{2^7}$? $2^2 = 4$. Letter E.

Let's look at Cell C again: $3^8 \cdot 3^2$. If it were $3^8 / 3^6$, it would be $3^2=9$ (J).

Let's look at the whole grid as a "Crossword" style math puzzle.

Let's try to interpret the blurry text more carefully based on standard algebra problems that yield small integers.

Row 1:
* C: $3^8 \cdot 3^{-6}$? If the second exponent is negative... $3^2 = 9$. J.
* Let's assume the dot is multiplication. If the second term is $3^{-6}$, it works. But it looks like $3^2$.
* What if it's $3^2 \cdot 3^2 = 3^4 = 81$? No.
* What if it's $\sqrt[3]{8}$? No.
* Let's skip and come back.
* Next box: $(8^2)^3$? $8^6$. Huge.
* Maybe it's $(8^{1/2})^3$? No.
* Maybe it's $\sqrt[3]{8}$? $2$. C.
* The text looks like `(8^?)^?`. It looks like `(8^1/3)`? If it's cube root of 8, that's 2.
* Next box: $10^4$. This is definitely 10,000. Unless it's $\log_{10}(10^4) = 4$?
* If we apply Log base 10 to everything?
* Cell W: $\log(3 \cdot 8) = \log 24 \approx 1.38$. Not an integer.

Let's look at Cell S: $\frac{6^5}{6}$.
$\frac{6^5}{6^1} = 6^4 = 1296$. Too big.

There is a possibility that the operations are Modulo 26?
$24 \pmod{26} = 24$ (Y).
$1296 \pmod{26}$: $1296 / 26 = 49.8$. $26 \times 50 = 1300$. $1300 - 4 = 1296$. So $1296 \equiv -4 \equiv 22$. W.

Let's test this "Modulo 26" hypothesis on a few clear ones.

Cell P: $\frac{7^8}{7^7} = 7^1 = 7$.
$7 \pmod{26} = 7$. H.

Cell R: $(5 \cdot 4)^2$? Or $(5 \cdot 4)^{-2}$?
Image looks like $(5 \cdot 4)^2 = 20^2 = 400$.
$400 \pmod{26}$: $26 \times 10 = 260$. $400 - 260 = 140$. $26 \times 5 = 130$. $140 - 130 = 10$.
$10 \rightarrow$ K.

Cell L: $(10^7)$?
$10^7 = 10,000,000$.
$10^7 \pmod{26}$.
$10^1 = 10$.
$10^2 = 100 = 26 \times 3 + 22 \equiv -4$.
$10^3 \equiv -40 \equiv -14 \equiv 12$.
$10^4 \equiv 120 = 26 \times 4 + 16 \equiv 16$.
$10^5 \equiv 160 = 26 \times 6 + 4 \equiv 4$.
$10^6 \equiv 40 \equiv 14$.
$10^7 \equiv 140 \equiv 10$.
$10 \rightarrow$ K.

Cell M: $\left(\frac{9^{12}}{9^?}\right)$? Looks like $\left(\frac{9^{12}}{9^9}\right)^2$?
Inside: $9^3 = 729$.
$729^2$?
Or maybe just $9^{12}/9^{10} = 9^2 = 81$?
$81 \pmod{26}$: $26 \times 3 = 78$. $81 - 78 = 3$.
$3 \rightarrow$ D.

Let's look at Cell V (bottom left): $6 \cdot 6^4 = 6^5 = 7776$.
$7776 \pmod{26}$.
$6^1 = 6$.
$6^2 = 36 \equiv 10$.
$6^3 \equiv 60 \equiv 8$.
$6^4 \equiv 48 \equiv 22 \equiv -4$.
$6^5 \equiv -24 \equiv 2$.
$2 \rightarrow$ C.

Cell N (next to V): $\frac{1}{3^{-3}} = 3^3 = 27$.
$27 \pmod{26} = 1$.
$1 \rightarrow$ B.

This Modulo 26 theory seems very promising because standard exponent rules produce huge numbers, but modulo 26 maps them to the alphabet range. Let's proceed with calculating every cell using Result mod 26.

---

### Step-by-Step Calculation (Mod 26)

Grid Positions & Labels:

Row 1:
1. C: $3^8 \cdot 3^2 = 3^{10}$.
$3^1=3, 3^2=9, 3^3=27\equiv1$.
Since $3^3 \equiv 1$, then $3^{10} = (3^3)^3 \cdot 3^1 \equiv 1^3 \cdot 3 = 3$.
Value: 3. Letter: D.

2. (Unlabeled, between C and T): The label is actually above the box? No, the letters are inside the boxes or next to them.
Let's map letters to boxes carefully.
Box 1 (Top Left): Label C. Expression: $3^8 \cdot 3^2$. Result: 3 (D).
Box 2: Label (blank?) or is the label T for the whole row? No, T is far right.
Let's look at the labels inside/next to boxes.
Box 1: C.
Box 2: Expression $(8^2)^3$? No, looks like $(8^{1/3})$? Or $\sqrt[3]{8}$?
Let's look at the text: `(8^?)^?`. It looks like `(8^1/3)` is unlikely notation.
Maybe it is $(8^2)/8$? $8^1=8$.
Maybe it is $8^{2/3}$? $(\sqrt[3]{8})^2 = 2^2 = 4$.
Let's assume the expression is $\sqrt[3]{8^2}$ or similar that yields a small number.
Actually, looking at crop 1, it says `(8^2)^3`? No, the exponent 3 is small.
Let's look at the label. The label W is below Box 1.
The label F is below Box 2? No, F is to the right of Box 2.

Let's restart the mapping based on visual position.

Box 1 (Top Left): $3^8 \cdot 3^2$. Label C is top right corner of box? Or is C the answer slot?
Usually, the letter *is* the variable name. So Box 1 is variable C.
Value of C = 3 -> D.

Box 2: Expression $(8^?)^?$. It looks like $(8^{1/3})$? No.
It looks like $(8^2)^{1/3}$? $8^{2/3} = 4$.
Or maybe it's just $8$?
Let's look at the label. There is no letter inside. But there is a T at the end of the row.
Wait, the letters A-Z are scattered around.
Let's identify which letter belongs to which expression.

* C: $3^8 \cdot 3^2$
* W: $(3 \cdot 8)$ -- This is clearly labeled W.
* U: $2^7 \cdot 2^3$ -- Labeled U.
* A: $\binom{4}{5}$? No, looks like $(\frac{4}{9})^5$? Or maybe $4^5 / 9^5$?
Let's look at Crop 4. It shows `(4/9)^5`? No, it shows `(4 over 9)`. And a 5?
Actually, it looks like $\left(\frac{4}{9}\right)$ is not right.
It looks like $4^5$? No.
Let's look at the label A. It is under the box with $\left(\frac{4}{9}\right)$?

Let's try to read the letters associated with each box directly from the image layout.

Row 1:
* Box 1: $3^8 \cdot 3^2$. Label C is near it.
* Box 2: $(8^2)^3$? Label? There is a T at the far right.
* Box 3: $10^4$.
* Box 4: $(\frac{2^9}{2})^4$?

This is confusing. Let's look at the letters placed *between* or *under* boxes.

* W is under Box 1 ($3^8 \cdot 3^2$?? No, W is under the box with $(3 \cdot 8)$).
Ah! The grid lines are tricky.

Let's trace the boxes properly.

Box 1 (Top Left): Contains $3^8 \cdot 3^2$. Label C is top-right.
Box 2: Contains $(8^2)^3$? No, looks like $(8^2)^3$ is wrong. It looks like $(8^{\frac{1}{3}})$?
Let's assume the standard simplification first without Mod 26 to see if any are naturally small.

If I don't use Mod 26, most are huge.
However, look at W: $(3 \cdot 8) = 24$. Naturally small.
Look at P: $7^8 / 7^7 = 7$. Naturally small.
Look at S: $6^5 / 6 = 6^4 = 1296$. Large.
Look at I: $6^5 / 6$? No, I is $\frac{6^5}{6}$?
Let's check the label I. It is next to $\frac{6^5}{6}$?
In Crop 2, we see $\frac{10^5}{10^3}$ labeled G?
$\frac{10^5}{10^3} = 10^2 = 100$.
$100 \pmod{26} = 22$ (W).

In Crop 2, we see $9^9 \cdot 9^{10}$? No, $9^9 \cdot 9^{-10}$?
It says $9^9 \cdot 9^{10}$? That's $9^{19}$.

Let's look at Crop 5 (Bottom Right).
Label M: $\left(\frac{9^{12}}{9^?}\right)^2$?
Denominator looks like $9^9$? Or $9^7$?
If it's $9^{12}/9^{10} = 9^2 = 81$. $81 \equiv 3$ (D).
If it's $9^{12}/9^9 = 9^3 = 729$. $729 = 26 \times 28 + 1$. $1$ (B).

Let's look at Crop 6 (Bottom Left).
Label V? No, label is missing or cut off.
Expression: $6 \cdot 6^4 = 6^5$.
Label L? No, L is $(10^7)$.

Okay, let's list the expressions and their likely labels based on proximity in the original image.

Row 1:
1. Expr: $3^8 \cdot 3^2$. Label: C.
Val: $3^{10}$. $3^3=27\equiv1$. $3^{10}=(3^3)^3 \cdot 3 \equiv 3$.
Letter: D.

2. Expr: $(8^2)^3$? Or $(8^{1/3})$?
Looking at the superscript, it looks like a `1/3` or just `3`?
Let's look at the label. There is no letter immediately attached.
Wait, the letter T is at the end of the row.
The letter F is between Row 1 and 2?

Let's look at the labels explicitly printed in the cells or corners.

* Cell (1,1): $3^8 \cdot 3^2$. Corner label C.
* Cell (1,2): $(8^2)^3$? Corner label? None visible inside. But F is to the right.
* Cell (1,3): $10^4$.
* Cell (1,4): $(\frac{2^9}{2})^4$? Corner label T.

* Cell (2,1): $(3 \cdot 8)$. Corner label W.
* Cell (2,2): $(4^1)^3$? Corner label F?
* Cell (2,3): $9^9 \cdot 9^{10}$? Corner label G?
* Cell (2,4): $\frac{10^5}{10^3}$. Corner label G is shared? Or I?
Actually, looking at Crop 2, G is near $\frac{10^5}{10^3}$.

* Cell (3,1): $2^7 \cdot 2^3$. Corner label U.
* Cell (3,2): $\frac{4^8}{5} \cdot 3$? No, $\frac{4^8}{5^3}$?
Label B.
* Cell (3,3): $(5 \cdot 4)^2$? Label Y.
* Cell (3,4): $\frac{6^5}{6}$. Label I? Or S?
Crop 2 shows I near $\frac{6^5}{6}$? No, I is top right of that cell. S is bottom right.

* Cell (4,1): $\binom{4}{5}$? No, $(\frac{4}{9})^5$?
Label A.
* Cell (4,2): $\frac{7^8}{7^7}$. Label P.
* Cell (4,3): $(5^2)^6 \cdot 5$? No, $(5^2)^6 \cdot 5^{-1}$?
Label R.
* Cell (4,4): $12^{25} \cdot 12^{-3}$?
Label ?

* Cell (5,1): $6 \cdot 6^4$. Label ? (Maybe V?)
* Cell (5,2): $\frac{1}{3^{-3}}$. Label ?
* Cell (5,3): $(10^7)$. Label L.
* Cell (5,4): $(\frac{9^{12}}{9^9})^2$? Label M.

Let's refine the calculations with the Mod 26 assumption, which is the only way these exponents make sense for a cipher.

1. Label C: $3^8 \cdot 3^2 = 3^{10}$
$3^1=3, 3^2=9, 3^3=27\equiv1$.
$3^{10} = (3^3)^3 \cdot 3^1 \equiv 1 \cdot 3 = 3$.
$3 \rightarrow$ D

2. Label W: $(3 \cdot 8) = 24$
$24 \rightarrow$ Y

3. Label U: $2^7 \cdot 2^3 = 2^{10}$
$2^{10} = 1024$.
$1024 / 26$: $26 \times 39 = 1014$.
$1024 - 1014 = 10$.
$10 \rightarrow$ K

4. Label A: The expression is $\left(\frac{4}{9}\right)^5$?
Fractions in modular arithmetic require modular inverses.
$4 \cdot 9^{-1} \pmod{26}$.
Does 9 have an inverse mod 26? $\gcd(9,26)=1$. Yes.
$9 \times 3 = 27 \equiv 1$. So $9^{-1} \equiv 3$.
Base: $4 \cdot 3 = 12$.
Expr: $12^5$.
$12^2 = 144 = 26 \times 5 + 14 \equiv 14$.
$12^4 \equiv 14^2 = 196 = 26 \times 7 + 14 \equiv 14$.
$12^5 = 12^4 \cdot 12 \equiv 14 \cdot 12 = 168$.
$168 / 26$: $26 \times 6 = 156$.
$168 - 156 = 12$.
$12 \rightarrow$ M

5. Label V (Bottom Left, $6 \cdot 6^4$):
$6^5$.
$6^1=6, 6^2=36\equiv10, 6^3=60\equiv8, 6^4=48\equiv22\equiv-4, 6^5=-24\equiv2$.
$2 \rightarrow$ C
*(Note: The label might be V, but let's check the position. The letter V is not explicitly visible, but it follows U. Let's assume the order is reading left-to-right, top-to-bottom of the labels? No, the labels are scattered. We must read the final message by assembling the letters in the order of the labels A, B, C... Z? Or by reading the grid rows?)*

Usually, these puzzles spell a word when read Left-to-Right, Top-to-Bottom regardless of the label letters. The label letters (A, B, C...) might just be identifiers for the boxes, or they might be the answer key indices.

Wait! The prompt says "Match that number to a letter using the provided key (A=0... Z=25)".
The letters IN the grid (C, W, U, A...) are likely the variables we are solving for.
The question is: What is the final message?
Is the final message the sequence of solved letters arranged by their position in the grid?
OR is the final message formed by ordering the variables A, B, C...?

Let's look at the blanks at the bottom:
`___ ___ ___ ___ ___ ___ ___`
`___ ___ ___ ___ ___ ___ __ !!`

This suggests a phrase of two words.

If we arrange the results by Grid Position (Row 1 Left to Right, etc.):

Row 1:
1. Box 1 (Label C): Result 3 -> D
2. Box 2 (Label ?): Expression $(8^2)^3$?
Let's look at Box 2 again. Crop 1.
It looks like $(8^{\frac{1}{3}})$? No, the 3 is high.
Maybe it is $8^{2/3}$?
$8^{2/3} = (\sqrt[3]{8})^2 = 2^2 = 4$.
$4 \rightarrow$ E
Let's assume the label is implicit or I missed it. Let's call this Box 2.
3. Box 3 (Label ?): $10^4$.
$10^4 = 10000$.
$10000 \pmod{26}$.
$10^2 = 100 \equiv 22 \equiv -4$.
$10^4 \equiv (-4)^2 = 16$.
$16 \rightarrow$ Q
4. Box 4 (Label T): $(\frac{2^9}{2})^4 = (2^8)^4 = 2^{32}$.
$2^{10} \equiv 10$.
$2^{20} \equiv 100 \equiv 22 \equiv -4$.
$2^{30} \equiv 2^{10} \cdot 2^{20} \equiv 10 \cdot (-4) = -40 \equiv 12$.
$2^{32} = 2^{30} \cdot 2^2 \equiv 12 \cdot 4 = 48 \equiv 22$.
$22 \rightarrow$ W

So Row 1 spells: D E Q W? Doesn't look like a word.

Let's re-read Box 2.
Maybe it's $(8^2)/8$? $8$. I.
Maybe it's $\sqrt[3]{8}$? $2$. C.

Let's look at the labels again.
The labels are: A, B, C, F, G, I, L, M, P, R, S, T, U, V(?), W, Y.

Maybe the message is formed by sorting the variables alphabetically?
A, B, C, ...

Let's solve all labeled variables.

A: $\left(\frac{4}{9}\right)^5 \equiv 12 \rightarrow$ M

B: Box with $\frac{4^8}{5} \cdot 3$?
Crop 3 shows: $\frac{4^8}{5} \cdot 3$?
Actually, it looks like $\frac{4^8}{5^3}$?
Or maybe $\frac{4^3}{5}$?
Let's look at the exponents. Top: 8? Bottom: 5? Side: 3?
Let's assume it's $\frac{4^8}{5} \cdot 3$ is weird.
How about $4^3 / 5$?
Let's look at the visual again.
It looks like $\frac{4^{\text{something}}}{5} \cdot 3$.
Maybe it's $\frac{4^2}{5}$?

Let's skip B for a moment and do the clearer ones.

C: $3^{10} \equiv 3 \rightarrow$ D

F: Box with $(4^1)^3$?
Crop 2 shows $(4^1)^3$.
$4^3 = 64$.
$64 = 26 \times 2 + 12$.
$12 \rightarrow$ M

G: Box with $\frac{10^5}{10^3}$?
Crop 2 shows $\frac{10^5}{10^3} = 10^2 = 100$.
$100 \equiv 22 \rightarrow$ W

I: Box with $\frac{6^5}{6}$?
Crop 2 shows $\frac{6^5}{6} = 6^4 = 1296$.
$1296 \equiv 22 \rightarrow$ W
*(Wait, earlier I calculated $6^5 \equiv 2$. Here it is $6^4$.)*
$6^2 = 36 \equiv 10$.
$6^4 \equiv 100 \equiv 22$.
So I = 22 -> W.

L: $(10^7)$.
$10^7 \equiv 10 \rightarrow$ K (Calculated earlier: $10^7 \equiv 10$).

M: $\left(\frac{9^{12}}{9^9}\right)^2$?
Crop 5: Numerator $9^{12}$, Denom $9^9$? Or $9^7$?
If $9^9$: Inside $9^3 = 729 \equiv 1$. $1^2 = 1 \rightarrow$ B.
If $9^7$: Inside $9^2 = 81 \equiv 3$. $3^2 = 9 \rightarrow$ J.
Visually, the denominator exponent looks like a 9 or a 7. It has a loop. Likely 9.
Let's assume B.

P: $\frac{7^8}{7^7} = 7^1 = 7 \rightarrow$ H

R: $(5^2)^6 \cdot 5$?
Crop 5: $(5^2)^6 \cdot 5$? Or $(5^2)^6 \cdot 5^{-1}$?
It looks like $(5^2)^6 \cdot 5$.
$5^{12} \cdot 5^1 = 5^{13}$.
Powers of 5 mod 26:
$5^1 = 5$.
$5^2 = 25 \equiv -1$.
$5^{12} = (5^2)^6 \equiv (-1)^6 = 1$.
$5^{13} = 1 \cdot 5 = 5$.
$5 \rightarrow$ F

S: Label S is near $\frac{6^5}{6}$?
In Crop 2, I is top right, S is bottom right of the same box?
Or is S the next box?
Crop 2 shows $\frac{6^5}{6}$ and label I and S.
Maybe the box is for I and S? Unlikely.
Let's look at Row 3, Box 4.
Label I is top right. Label S is bottom right.
Maybe the expression is for S?
If S is $\frac{6^5}{6} = 6^4 \equiv 22 \rightarrow$ W.

T: $(\frac{2^9}{2})^4 = 2^{32} \equiv 22 \rightarrow$ W

U: $2^7 \cdot 2^3 = 2^{10} \equiv 10 \rightarrow$ K

V: (Bottom Left, $6 \cdot 6^4 = 6^5$).
$6^5 \equiv 2 \rightarrow$ C

W: $(3 \cdot 8) = 24 \rightarrow$ Y

Y: Box with $(5 \cdot 4)^2$?
Crop 3: $(5 \cdot 4)^2 = 20^2 = 400$.
$400 \equiv 10 \rightarrow$ K

Missing Labels/Boxes:

Box 2 (Row 1): $(8^2)^3$?
If it's $8^6$:
$8^1=8, 8^2=64\equiv12, 8^3=96\equiv18, 8^4=144\equiv14, 8^5=112\equiv8, 8^6=64\equiv12$.
$12 \rightarrow$ M.

Box 3 (Row 1): $10^4 \equiv 16 \rightarrow$ Q.

Box 2 (Row 2): $(4^1)^3 = 64 \equiv 12 \rightarrow$ M. (Label F?)

Box 3 (Row 2): $9^9 \cdot 9^{10}$?
$9^{19}$.
$9^1=9, 9^2=81\equiv3, 9^3=27\equiv1$.
$9^{19} = (9^3)^6 \cdot 9^1 \equiv 1 \cdot 9 = 9$.
$9 \rightarrow$ J. (Label G?)

Box 2 (Row 3): $\frac{4^8}{5} \cdot 3$?
Let's assume the expression is $\frac{4^3}{5}$?
$4^3 = 64 \equiv 12$.
$12 / 5 = 12 \cdot 5^{-1}$.
$5 \cdot ? \equiv 1 \pmod{26}$.
$5 \cdot 21 = 105 = 4 \times 26 + 1$. So $5^{-1} = 21 \equiv -5$.
$12 \cdot (-5) = -60 \equiv 18$.
$18 \rightarrow$ S.

Box 4 (Row 4): $12^{25} \cdot 12^{-3}$?
Crop 5: $12^{25} \cdot 12^{-3}$?
$12^{22}$.
$12^2 = 144 \equiv 14$.
$12^3 = 168 \equiv 12$.
$12^4 \equiv 14$.
Pattern: Even powers $\equiv 14$, Odd powers $\equiv 12$?
$12^1=12$.
$12^2=14$.
$12^3=12$.
$12^4=14$.
So $12^{22} \equiv 14$.
$14 \rightarrow$ O.

Box 2 (Row 5): $\frac{1}{3^{-3}} = 3^3 = 27 \equiv 1 \rightarrow$ B.

---

Constructing the Message:

If we read the grid Left-to-Right, Top-to-Bottom:

Row 1:
1. C: D
2. (No Label): M (Assuming $8^6$)
3. (No Label): Q ($10^4$)
4. T: W

Row 2:
1. W: Y
2. F: M
3. G: J
4. I/S?: W

Row 3:
1. U: K
2. B: S (Assuming $4^3/5$)
3. Y: K
4. ?: ?

This doesn't spell anything obvious.

Alternative Strategy:
Look at the blanks: `___ ___ ___ ___ ___ ___ ___` (7 letters)
`___ ___ ___ ___ ___ ___ __ !!` (7 letters + !!)

Total 14 letters.

Let's count the boxes.
Row 1: 4 boxes.
Row 2: 4 boxes.
Row 3: 4 boxes.
Row 4: 4 boxes.
Row 5: 4 boxes.
Total 20 boxes.

Maybe only specific boxes form the message?

Let's look at the labels that ARE letters of the alphabet.
A, B, C, F, G, I, L, M, P, R, S, T, U, W, Y.

If we sort the results by Label Name (A-Z):

A: M
B: S
C: D
F: M
G: J
I: W
L: K
M: B
P: H
R: F
S: W
T: W
U: K
W: Y
Y: K

Sequence: M S D M J W K B H F W W K Y K
Doesn't make sense.

Let's try reading the Grid Columns?

Let's try identifying the Word directly.
Common phrases in these puzzles: "GOOD JOB", "WELL DONE", "KEEP TRYING".

Let's check "WELL DONE".
W (24), E (4), L (11), L (11), D (3), O (14), N (13), E (4).

Do we have these values?
W=24 (Box W).
E=4 (Box 2 Row 1 if $8^{2/3}$).
L=11?
D=3 (Box C).

Let's check "GOOD WORK".

Let's look at the first row again: D M Q W.

What if the operations are NOT Mod 26?
What if they are just simple arithmetic and I'm misreading the exponents?

Box W: $3 \cdot 8 = 24$ (Y).
Box P: $7^8/7^7 = 7$ (H).

If W=Y and P=H...

Let's look at the bottom line blanks.
`_______ _______!!`

Maybe the answer is "YOU WIN"?
Y (24), O (14), U (20), W (22), I (8), N (13).

Let's check if we have Y, O, U, W, I, N.
Y: Box W gives 24 (Y).
O: Box ? gives 14?
Box 4 Row 4 gave 14 (O).
U: Box ? gives 20?
Box U gave 10 (K).

This is getting speculative.

Final Check of the most distinct boxes:

1. W = 24 -> Y
2. P = 7 -> H
3. C = 3 -> D
4. U = 10 -> K
5. L = 10 -> K
6. T = 22 -> W

Letters found: Y, H, D, K, K, W.

If we arrange