Question

practice: exponent rule 5 - power rule
on a piece of paper, try the problems below. more practice is provided at the end of the lesson (last slide).

1. $(4^{3})^{5}$
2. $(a^{6})^{4}$
3. $(2x^{2})^{3}$
4. $(5x^{2}y^{4})^{3}$
5. $(2x^{4})^{3}$
6. $(5a^{7}b^{6})^{3}$
7. $(x^{-4})^{-4}$
8. (crossed out content)

Explanation:

The Power Rule for exponents states that $(a^m)^n = a^{m \times n}$, and for a product raised to a power: $(ab)^n = a^n b^n$. We apply this to each problem:

Step1: Apply power rule to $(4^3)^5$

Multiply the exponents: $4^{3 \times 5} = 4^{15}$

Step2: Apply power rule to $(a^6)^4$

Multiply the exponents: $a^{6 \times 4} = a^{24}$

Step3: Apply power rule to $(2x^4)^3$

Raise each factor to the power: $2^3 \times x^{4 \times 3} = 8x^{12}$

Step4: Apply power rule to $(5a^7b^6)^3$

Raise each factor to the power: $5^3 \times a^{7 \times 3} \times b^{6 \times 3} = 125a^{21}b^{18}$

Step5: Apply power rule to $(2x^2)^3$

Raise each factor to the power: $2^3 \times x^{2 \times 3} = 8x^6$

Step6: Apply power rule to $(5x^2y^4)^3$

Raise each factor to the power: $5^3 \times x^{2 \times 3} \times y^{4 \times 3} = 125x^6y^{12}$

Step7: Apply power rule to $(x^{-4})^{-4}$

Multiply the exponents: $x^{(-4) \times (-4)} = x^{16}$

Answer:

1. $4^{15}$
2. $a^{24}$
3. $8x^{12}$
4. $125a^{21}b^{18}$
5. $8x^6$
6. $125x^6y^{12}$
7. $x^{16}$