It looks like you've uploaded an image of a
coloring activity worksheet focused on
exponential equations, created by
All Things Algebra®. While I can't see the full worksheet or all the problems, I can help solve the visible exponential equations and explain how to approach them.
Let’s go through the
visible equations from the image:
---
🔹 Problem 1:
$$
7^{4x - 9} = 7^{x + 6}
$$
####
✔ Solution:
Since the bases are the same (both are 7), we can set the exponents equal to each other:
$$
4x - 9 = x + 6
$$
Subtract $x$ from both sides:
$$
3x - 9 = 6
$$
Add 9 to both sides:
$$
3x = 15
$$
Divide by 3:
$$
x = 5
$$
✔ Answer: $x = 5$
---
🔹 Problem 3:
$$
6^x \cdot 6^{x+5} = 6^{5x+2} \cdot 6^{4x+10}
$$
####
✔ Solution:
Use the
product of powers rule: $a^m \cdot a^n = a^{m+n}$
Left side:
$$
6^x \cdot 6^{x+5} = 6^{x + (x+5)} = 6^{2x + 5}
$$
Right side:
$$
6^{5x+2} \cdot 6^{4x+10} = 6^{(5x+2) + (4x+10)} = 6^{9x + 12}
$$
Now set the exponents equal:
$$
2x + 5 = 9x + 12
$$
Subtract $2x$ from both sides:
$$
5 = 7x + 12
$$
Subtract 12:
$$
-7 = 7x
$$
Divide by 7:
$$
x = -1
$$
✔ Answer: $x = -1$
---
🔹 Problem 5:
$$
\frac{1}{10} = 10^{3x + 20}
$$
####
✔ Solution:
Note that $\frac{1}{10} = 10^{-1}$, so rewrite the equation:
$$
10^{-1} = 10^{3x + 20}
$$
Set the exponents equal:
$$
-1 = 3x + 20
$$
Subtract 20:
$$
-21 = 3x
$$
Divide by 3:
$$
x = -7
$$
✔ Answer: $x = -7$
---
🔹 Problem 7:
$$
32^{x - 1} = 8^{2x + 3}
$$
####
✔ Solution:
Express both sides with the
same base. Note:
- $32 = 2^5$
- $8 = 2^3$
Rewrite:
$$
(2^5)^{x - 1} = (2^3)^{2x + 3}
$$
Use power of a power: $(a^m)^n = a^{mn}$
Left:
$$
2^{5(x - 1)} = 2^{5x - 5}
$$
Right:
$$
2^{3(2x + 3)} = 2^{6x + 9}
$$
Set exponents equal:
$$
5x - 5 = 6x + 9
$$
Subtract $5x$:
$$
-5 = x + 9
$$
Subtract 9:
$$
x = -14
$$
✔ Answer: $x = -14$
---
🎨 Coloring Activity Explanation:
After solving each equation, students likely find the value of $x$, then use that to determine which color to apply to a section of the sun-shaped coloring page. Each number (like 1, 2, 3, etc.) corresponds to a specific color based on a key (not shown here).
For example:
- If $x = 5$, maybe they color sections labeled "5" with a certain color.
- The solved values (e.g., $x = 5$, $x = -1$, etc.) may be used to match numbers on the coloring page.
---
✔ Summary of Solutions:
| Problem | Equation | Answer |
|--------|---------|--------|
| 1 | $7^{4x - 9} = 7^{x + 6}$ | $x = 5$ |
| 3 | $6^x \cdot 6^{x+5} = 6^{5x+2} \cdot 6^{4x+10}$ | $x = -1$ |
| 5 | $\frac{1}{10} = 10^{3x + 20}$ | $x = -7$ |
| 7 | $32^{x - 1} = 8^{2x + 3}$ | $x = -14$ |
---
If you’d like, I can help create a
coloring key or explain how to use these answers in the activity! Just let me know what part you need next. 😊
Parent Tip: Review the logic above to help your child master the concept of solving exponential equations worksheet with answers.
