Exponents Worksheets with Answer Key - Free Printable

Let's solve each problem step by step from the Powers and Exponents Worksheet.

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1. Simplify

#### a) $\left(\frac{2}{3}\right)^9 \times \left(\frac{4}{9}\right)^6$

First, note that:
$$
\frac{4}{9} = \left(\frac{2}{3}\right)^2
$$

So:
$$
\left(\frac{4}{9}\right)^6 = \left(\left(\frac{2}{3}\right)^2\right)^6 = \left(\frac{2}{3}\right)^{12}
$$

Now:
$$
\left(\frac{2}{3}\right)^9 \times \left(\frac{2}{3}\right)^{12} = \left(\frac{2}{3}\right)^{9+12} = \left(\frac{2}{3}\right)^{21}
$$

✔ Answer: $\left(\frac{2}{3}\right)^{21}$

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#### b) $\frac{(3b^3)^9}{(9b^2)^4}$

Break it down:

Numerator:
$$
(3b^3)^9 = 3^9 \cdot (b^3)^9 = 3^9 \cdot b^{27}
$$

Denominator:
$$
(9b^2)^4 = (9)^4 \cdot (b^2)^4 = (3^2)^4 \cdot b^8 = 3^8 \cdot b^8
$$

Now divide:
$$
\frac{3^9 \cdot b^{27}}{3^8 \cdot b^8} = 3^{9-8} \cdot b^{27-8} = 3^1 \cdot b^{19} = 3b^{19}
$$

✔ Answer: $3b^{19}$

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#### c) $\frac{(4x^2)^6}{(2x^4)^4}$

Numerator:
$$
(4x^2)^6 = 4^6 \cdot (x^2)^6 = (2^2)^6 \cdot x^{12} = 2^{12} \cdot x^{12}
$$

Denominator:
$$
(2x^4)^4 = 2^4 \cdot (x^4)^4 = 2^4 \cdot x^{16}
$$

Now divide:
$$
\frac{2^{12} \cdot x^{12}}{2^4 \cdot x^{16}} = 2^{12-4} \cdot x^{12-16} = 2^8 \cdot x^{-4} = \frac{256}{x^4}
$$

✔ Answer: $\frac{256}{x^4}$

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#### d) $(-4)^{100} \times (-4)^{20}$

Use rule: $a^m \times a^n = a^{m+n}$

So:
$$
(-4)^{100 + 20} = (-4)^{120}
$$

Note: Since exponent is even, this is positive:
$$
(-4)^{120} = 4^{120}
$$

But unless asked to simplify further, we can leave as:
$$
(-4)^{120}
$$

✔ Answer: $(-4)^{120}$

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2. Find the value of $x$

#### a) $5^{(x/2)} = 5^x$

Since bases are same:
$$
\frac{x}{2} = x
\Rightarrow \frac{x}{2} - x = 0
\Rightarrow -\frac{x}{2} = 0
\Rightarrow x = 0
$$

✔ Answer: $x = 0$

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#### b) $\left(\frac{8}{9}\right)^5 \times \left(\frac{9}{4}\right) = 2x$

First compute left side.

Note:
$$
\frac{8}{9} = \frac{2^3}{3^2},\quad \frac{9}{4} = \frac{3^2}{2^2}
$$

So:
$$
\left(\frac{8}{9}\right)^5 = \frac{2^{15}}{3^{10}},\quad \text{then multiply by } \frac{9}{4} = \frac{3^2}{2^2}
$$

So total:
$$
\frac{2^{15}}{3^{10}} \times \frac{3^2}{2^2} = 2^{15-2} \cdot 3^{2-10} = 2^{13} \cdot 3^{-8}
$$

Now set equal to $2x$:
$$
2x = 2^{13} \cdot 3^{-8} \Rightarrow x = 2^{12} \cdot 3^{-8} = \frac{2^{12}}{3^8}
$$

We can compute values:
- $2^{12} = 4096$
- $3^8 = 6561$

So:
$$
x = \frac{4096}{6561}
$$

✔ Answer: $x = \frac{4096}{6561}$

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#### c) $6^x = 216$

Note: $216 = 6^3$, because:
$$
6^1 = 6,\quad 6^2 = 36,\quad 6^3 = 216
$$

So:
$$
6^x = 6^3 \Rightarrow x = 3
$$

✔ Answer: $x = 3$

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#### d) $x(3^{-5}) = 3$

Solve for $x$:
$$
x = \frac{3}{3^{-5}} = 3 \cdot 3^5 = 3^{1+5} = 3^6 = 729
$$

✔ Answer: $x = 729$

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3. Write in exponential form

#### a) $\left(\frac{2}{5}\right)^{-2} \times \left(\frac{2}{5}\right)^{-2} \times \left(\frac{2}{5}\right)^{-2}$

This is three identical terms multiplied:
$$
= \left(\frac{2}{5}\right)^{-2 + (-2) + (-2)} = \left(\frac{2}{5}\right)^{-6}
$$

✔ Answer: $\left(\frac{2}{5}\right)^{-6}$

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#### b) $\left(\frac{5}{2}\right)^{-1} \times \left(\frac{5}{2}\right)^{-1} \times \left(\frac{5}{2}\right)^{-1} \times \left(\frac{5}{2}\right)^{-1}$

Four times:
$$
= \left(\frac{5}{2}\right)^{-1 \times 4} = \left(\frac{5}{2}\right)^{-4}
$$

✔ Answer: $\left(\frac{5}{2}\right)^{-4}$

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4. Find the values of the following

#### a) $(3^{-1} + 4^{-1} + 5^{-1})^0$

Any non-zero number raised to power 0 is 1.

Check if inside is zero:
- $3^{-1} = \frac{1}{3},\quad 4^{-1} = \frac{1}{4},\quad 5^{-1} = \frac{1}{5}$
- Sum: $\frac{1}{3} + \frac{1}{4} + \frac{1}{5} = \frac{20 + 15 + 12}{60} = \frac{47}{60} \ne 0$

So:
$$
(\text{non-zero})^0 = 1
$$

✔ Answer: $1$

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#### b) $(3^0 + 4^{-1}) \times 2^2$

Compute each part:
- $3^0 = 1$
- $4^{-1} = \frac{1}{4}$
- So: $1 + \frac{1}{4} = \frac{5}{4}$
- $2^2 = 4$

Now:
$$
\frac{5}{4} \times 4 = 5
$$

✔ Answer: $5$

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✔ Final Answers Summary:

#### 1. Simplify
a) $\left(\frac{2}{3}\right)^{21}$
b) $3b^{19}$
c) $\frac{256}{x^4}$
d) $(-4)^{120}$

#### 2. Find $x$
a) $x = 0$
b) $x = \frac{4096}{6561}$
c) $x = 3$
d) $x = 729$

#### 3. Exponential Form
a) $\left(\frac{2}{5}\right)^{-6}$
b) $\left(\frac{5}{2}\right)^{-4}$

#### 4. Find Values
a) $1$
b) $5$

Let me know if you'd like these explained in more detail!