Sure! Let's tackle each part of the problem step-by-step.
### Part (b) Simplify [tex]\(\left(\frac{7}{x}\right)^{-3}\)[/tex]
1. Understanding negative exponents: The property of negative exponents states that [tex]\(a^{-n} = \frac{1}{a^n}\)[/tex]. Thus, for any expression of the form [tex]\(\left(\frac{a}{b}\right)^{-n}\)[/tex], we can write it as [tex]\(\left(\frac{b}{a}\right)^n\)[/tex].
2. Apply the property: For the given expression [tex]\(\left(\frac{7}{x}\right)^{-3}\)[/tex], using the negative exponent rule:
[tex]\[
\left(\frac{7}{x}\right)^{-3} = \left(\frac{x}{7}\right)^3
\][/tex]
3. Simplify the expression: Next, we can use the power rule which states [tex]\(\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\)[/tex]:
[tex]\[
\left(\frac{x}{7}\right)^3 = \frac{x^3}{7^3}
\][/tex]
4. Evaluate the constants: Calculate [tex]\(7^3\)[/tex]:
[tex]\[
7^3 = 343
\][/tex]
Therefore, the simplified expression is:
[tex]\[
\left(\frac{7}{x}\right)^{-3} = \frac{x^3}{343}
\][/tex]
### Part (c) Solve [tex]\(a^8 \div a^p = a^2\)[/tex] for [tex]\(p\)[/tex]
1. Rewrite the division using exponent rules: The division of exponents with the same base can be written as the subtraction of the exponents:
[tex]\[
a^8 \div a^p = a^{8-p}
\][/tex]
2. Set the expression equal to the given result: According to the problem, this is equal to [tex]\(a^2\)[/tex]. Therefore, we have:
[tex]\[
a^{8-p} = a^2
\][/tex]
3. Equate the exponents: Since the bases are the same and non-zero, the exponents must be equal:
[tex]\[
8 - p = 2
\][/tex]
4. Solve for [tex]\(p\)[/tex]:
[tex]\[
8 - 2 = p
\][/tex]
[tex]\[
p = 6
\][/tex]
Therefore, the value of [tex]\(p\)[/tex] is [tex]\(6\)[/tex].
In summary:
- For part (b), the simplified form of [tex]\(\left(\frac{7}{x}\right)^{-3}\)[/tex] is [tex]\(\frac{x^3}{343}\)[/tex].
- For part (c), the value of [tex]\(p\)[/tex] that satisfies [tex]\(a^8 \div a^p = a^2\)[/tex] is [tex]\(p = 6\)[/tex].
