High School

23. Reduce:

P = \frac{\left[x^9 \cdot (x^5)^{-3}\right]^{-3}}{x^{-2} \cdot x^{4^2}}

a) x^{13}

b) x^8

c) x^7

d) x^4

e) x^{10}

Answer :

To solve the expression [tex]P = \frac{\left[x^9 \cdot (x^5)^{-3}\right]^{-3}}{x^{-2} \cdot x^{4^2}}[/tex] and reduce it to one of the given options, we need to follow the laws of exponents. Let's break it down step-by-step:


  1. Simplify the numerator:

    [tex]\left[x^9 \cdot (x^5)^{-3}\right]^{-3}[/tex]

    First, simplify [tex](x^5)^{-3}[/tex]. This is equivalent to [tex]x^{-15}[/tex] because when you raise a power to another power, you multiply the exponents: [tex]5 \times -3 = -15[/tex].

    So, the expression becomes:
    [tex]\left[x^9 \cdot x^{-15}\right]^{-3}[/tex]

    Combine the exponents in [tex]x^9 \cdot x^{-15}[/tex]:
    [tex]x^{9-15} = x^{-6}[/tex].

    Now, apply the power of [tex]-3[/tex]:
    [tex]\left[ x^{-6} \right]^{-3} = x^{18}[/tex]
    Because [tex](-6) \times (-3) = 18[/tex].


  2. Simplify the denominator:

    [tex]x^{-2} \cdot x^{4^2}[/tex]

    Calculating [tex]4^2 = 16[/tex], the expression becomes:
    [tex]x^{-2} \cdot x^{16}[/tex]

    Combine the exponents in the denominator:
    [tex]x^{-2 + 16} = x^{14}[/tex].


  3. Combine the simplified numerator and denominator:

    Now, combine the parts:
    [tex]\frac{x^{18}}{x^{14}}[/tex]

    Use the property of division to subtract the exponents:
    [tex]x^{18-14} = x^{4}[/tex]



The final reduced expression is [tex]x^4[/tex].

Therefore, the correct answer is (d) [tex]x^4[/tex].

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