Question

question 5 of 9 (1) rewrite the following without an exponent: \\(\frac{1}{2^{-4}}\\)
question 6 of 9 (1) rewrite the expression without using a negative exponent: \\(6p^{-2}\\) simplify your answer as much as possible.
question 7 of 9 (2) simplify. \\(2v^{5}\cdot 4x^{5}\cdot 2w^{-7}x^{-5}w^{4}v\\) use only positive exponents in your answer.
question 8 of 9 (1) write the following as a radical expression. \\(5^{\frac{1}{6}}\\)
question 9 of 9 (1) simplify. \\(y^{\frac{3}{4}}\cdot y^{\frac{7}{8}}\\) assume that the variable represents a positive real number.

Explanation:

Question 5

Step1: Apply negative exponent rule

Recall $a^{-n}=\frac{1}{a^n}$, so $2^{-4}=\frac{1}{2^4}$.
$\frac{1}{2^{-4}} = 2^4$

Step2: Calculate the power

$2^4=2\times2\times2\times2$
$2^4=16$

Question 6

Step1: Apply negative exponent rule

$p^{-2}=\frac{1}{p^2}$
$6p^{-2}=6\times\frac{1}{p^2}$

Step2: Simplify the expression

$\frac{6}{p^2}$

Question 7

Step1: Group like terms

Group constants, $v$, $x$, $w$ terms:
$(2\times4\times2) \cdot (v^5 \cdot v) \cdot (x^5 \cdot x^{-5}) \cdot (w^{-7} \cdot w^4)$

Step2: Multiply constants

$2\times4\times2=16$

Step3: Combine $v$ terms (add exponents)

$v^{5+1}=v^6$

Step4: Combine $x$ terms (add exponents)

$x^{5+(-5)}=x^0=1$

Step5: Combine $w$ terms (add exponents)

$w^{-7+4}=w^{-3}=\frac{1}{w^3}$

Step6: Multiply all simplified terms

$16 \cdot v^6 \cdot 1 \cdot \frac{1}{w^3}$

Question 8

Step1: Apply rational exponent rule

Recall $a^{\frac{1}{n}}=\sqrt[n]{a}$, so $5^{\frac{1}{6}}=\sqrt[6]{5}$

Question 9

Step1: Apply exponent addition rule

For same base, $a^m \cdot a^n=a^{m+n}$:
$y^{\frac{3}{4}+\frac{7}{8}}$

Step2: Find common denominator and add

$\frac{3}{4}=\frac{6}{8}$, so $\frac{6}{8}+\frac{7}{8}=\frac{13}{8}$
$y^{\frac{13}{8}}$

Answer:

1. Question 5: $16$
2. Question 6: $\frac{6}{p^2}$
3. Question 7: $\frac{16v^6}{w^3}$
4. Question 8: $\sqrt[6]{5}$
5. Question 9: $y^{\frac{13}{8}}$