Question

1. for what value of p is the given expression equivalent to ( x^{29} ) when ( x

eq 0 )?
( \frac{(x^{3})^{p}}{(x^{4})^{-2}} )
type a response

Explanation:

Step1: Simplify the numerator

Using the power of a power rule \((a^m)^n = a^{mn}\), for \((x^3)^p\), we get \(x^{3p}\).

Step2: Simplify the denominator

Using the power of a power rule \((a^m)^n = a^{mn}\), for \((x^4)^{-2}\), we have \(x^{4\times(-2)} = x^{-8}\).

Step3: Simplify the fraction

Using the rule \(\frac{a^m}{a^n}=a^{m - n}\), the expression \(\frac{(x^3)^p}{(x^4)^{-2}}\) becomes \(x^{3p-(-8)}=x^{3p + 8}\).

Step4: Solve for \(p\)

Since the expression is equivalent to \(x^{29}\), we set the exponents equal: \(3p+8 = 29\).
Subtract 8 from both sides: \(3p=29 - 8=21\).
Divide both sides by 3: \(p=\frac{21}{3}=7\).

Answer:

7