Power of a Power

Show your work for each problem.

1.
Simplify $(x^3)^4$ .
1. a) $x^{7}$
2. b) $x^{81}$
3. c) $x^{12}$
4. d) $4x^{3}$
Answer: $x^{12}$
$(x^3)^4 = x^{3 \cdot 4} = x^{12}$ . The answer $x^7$ comes from adding the exponents, which is the rule for $x^3 \cdot x^4$ .
2.
$(2^3)^2 = 2^{\square}$ . What number goes in the box?
Answer: 6
$(2^3)^2 = 2^{3 \cdot 2} = 2^6$ . Check: $8^2 = 64$ and $2^6 = 64$ .
3.
Simplify $(a^5)^2$ by writing it as a product first.
Answer: a^10
1. How many factors of $a^5$ are in $(a^5)^2$ ? 2
2. $a^5 \cdot a^5 = a^{?}$ (add the exponents) 10
3. Check with the shortcut: $5 \cdot 2 = ?$ 10
4.
Evaluate $(3^2)^2$ .
Answer: 81
$(3^2)^2 = 9^2 = 81$ . With the rule: $3^{2 \cdot 2} = 3^4 = 81$ .
5.
Which expression is equal to $(5y)^2$ ?
1. a) $25y^2$
2. b) $25y$
3. c) $10y^2$
4. d) $5y^2$
Answer: $25y^2$
$(5y)^2 = 5^2 \cdot y^2 = 25y^2$ . $5y^2$ forgets to square the $5$ , and $10y^2$ multiplies $5$ by $2$ instead of squaring it.
6.
Compare $(2^2)^3$ and $(2^3)^2$ .
Answer: 64
1. $(2^2)^3 = 2^{?}$ 6
2. $(2^3)^2 = 2^{?}$ 6
3. Both are $2^6$ . What is $2^6$ ? 64
7.
$(2x^3)^4 = \square \, x^{12}$ . What number goes in the box?
Answer: 16
Every factor gets the exponent: $(2x^3)^4 = 2^4 \cdot (x^3)^4 = 16x^{12}$ .
8.
What is the value of $2^{3^2}$ (without brackets)?
1. a)512
2. b)128
3. c)36
4. d)64
Answer: 512
Without brackets the top exponent comes first: $3^2 = 9$ , so $2^{3^2} = 2^9 = 512$ . The value $64$ belongs to $(2^3)^2 = 2^6$ .
9.
Simplify $(3x^2)^3$ .
Answer: 27x^6
1. Raise the coefficient: $3^3 = ?$ 27
2. $(x^2)^3 = x^{?}$ 6
3. Check with $x = 1$ : $(3 \cdot 1^2)^3 = ?$ 27
10.
Simplify $\left(\frac{x^2}{3}\right)^3$ .
Answer: x^6/27
1. Numerator: $(x^2)^3 = x^{?}$ 6
2. Denominator: $3^3 = ?$ 27
3. Check with $x = 3$ : $\frac{3^6}{27} = \frac{729}{27} = ?$ 27
11.
$(10^2)^3 \cdot 10^4 = 10^{\square}$ . What number goes in the box?
Answer: 10
$(10^2)^3 = 10^{2 \cdot 3} = 10^6$ , and $10^6 \cdot 10^4 = 10^{6+4} = 10^{10}$ .
12.
Which statement is true for all numbers $a$ and $b$ ?
1. a) $(2a)^3 = 2a^3$
2. b) $(a + b)^2 = a^2 + b^2$
3. c) $(ab)^2 = a^2 b^2$
4. d) $(a^2)^3 = a^5$
Answer: $(ab)^2 = a^2 b^2$
Only $(ab)^2 = a^2 b^2$ is always true. In fact $(a + b)^2 = a^2 + 2ab + b^2$ , $(a^2)^3 = a^6$ and $(2a)^3 = 8a^3$ .
13.
Show with numbers that $(3 + 4)^2 \neq 3^2 + 4^2$ .
Answer: 24
1. $(3 + 4)^2 = 7^2 = ?$ 49
2. $3^2 + 4^2 = ?$ 25
3. How much is missing? $49 - 25 = ?$ 24
14.
Simplify $\frac{(x^4)^3 \cdot x^2}{(x^3)^4}$ .
Answer: x^2
1. $(x^4)^3 = x^{?}$ 12
2. $x^{12} \cdot x^2 = x^{?}$ 14
3. $(x^3)^4 = x^{?}$ 12
4. $\frac{x^{14}}{x^{12}} = x^{?}$ 2
15.
$(2a^2b^3)^4 = 16a^8 b^{\square}$ . What number goes in the box?
Answer: 12
$(2a^2b^3)^4 = 2^4 \cdot a^{2 \cdot 4} \cdot b^{3 \cdot 4} = 16a^8b^{12}$ . Each of the three factors is raised to the power.
16.
Simplify $(2x^2y)^3 \cdot (xy^2)^2$ .
Answer: 8x^8y^7
1. Coefficient of $(2x^2y)^3$ : $2^3 = ?$ 8
2. Exponent of $x$ in $(2x^2y)^3$ 6
3. Exponent of $y$ in $(xy^2)^2$ 4
4. Total exponent of $x$ after multiplying: $x^6 \cdot x^2 = x^{?}$ 8
5. Total exponent of $y$ after multiplying: $y^3 \cdot y^4 = y^{?}$ 7
17.
Simplify $\left(\frac{2x^3}{y^2}\right)^3$ .
1. a) $\frac{6x^9}{y^6}$
2. b) $\frac{2x^9}{y^6}$
3. c) $\frac{8x^6}{y^5}$
4. d) $\frac{8x^9}{y^6}$
Answer: $\frac{8x^9}{y^6}$
$\left(\frac{2x^3}{y^2}\right)^3 = \frac{2^3 \cdot x^{3 \cdot 3}}{y^{2 \cdot 3}} = \frac{8x^9}{y^6}$ . $6x^9$ multiplies $2$ by $3$ , $x^6$ and $y^5$ add the exponents, and $2x^9$ forgets the coefficient.
18.
A cube has edges of length $3 \cdot 10^2$ cm, so its volume is $(3 \cdot 10^2)^3$ cm³. Find the volume.
Answer: 27000000
1. $3^3 = ?$ 27
2. $(10^2)^3 = 10^{?}$ 6
3. Write the volume $27 \cdot 10^6$ cm³ as an ordinary number (digits only) 27000000