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Question20) Simplify:- (3−2)2×(52)−3×(t−3)2(3−2)5×(53)−2×(t−4)3\frac{\left(3^{-2}\right)^{2} \times\left(5^{2}\right)^{-3} \times\left(t^{-3}\right)^{2}}{\left(3^{-2}\right)^{5} \times\left(5^{3}\right)^{-2} \times\left(t^{-4}\right)^{3}}

Studdy Solution

STEP 1

1. We are asked to simplify the given expression.
2. The expression involves exponents and can be simplified using the laws of exponents.

STEP 2

1. Apply the power of a power rule to simplify each term in the numerator and the denominator.
2. Simplify the expression by combining like bases using the laws of exponents.
3. Simplify the fraction by canceling common factors.

STEP 3

Apply the power of a power rule (am)n=am⋅n(a^m)^n = a^{m \cdot n} to each term in the numerator:
(3−2)2=3−4,(52)−3=5−6,(t−3)2=t−6\left(3^{-2}\right)^{2} = 3^{-4}, \quad \left(5^{2}\right)^{-3} = 5^{-6}, \quad \left(t^{-3}\right)^{2} = t^{-6}
The numerator simplifies to:
3−4×5−6×t−63^{-4} \times 5^{-6} \times t^{-6}

STEP 4

Apply the power of a power rule to each term in the denominator:
(3−2)5=3−10,(53)−2=5−6,(t−4)3=t−12\left(3^{-2}\right)^{5} = 3^{-10}, \quad \left(5^{3}\right)^{-2} = 5^{-6}, \quad \left(t^{-4}\right)^{3} = t^{-12}
The denominator simplifies to:
3−10×5−6×t−123^{-10} \times 5^{-6} \times t^{-12}

STEP 5

Combine like bases in the fraction using the quotient of powers rule aman=am−n\frac{a^m}{a^n} = a^{m-n}:
For base 33:
3−43−10=3−4−(−10)=36\frac{3^{-4}}{3^{-10}} = 3^{-4 - (-10)} = 3^{6}
For base 55:
5−65−6=5−6−(−6)=50=1\frac{5^{-6}}{5^{-6}} = 5^{-6 - (-6)} = 5^{0} = 1
For base tt:
t−6t−12=t−6−(−12)=t6\frac{t^{-6}}{t^{-12}} = t^{-6 - (-12)} = t^{6}

STEP 6

Combine the simplified terms:
36×1×t6=36×t63^{6} \times 1 \times t^{6} = 3^{6} \times t^{6}
The simplified expression is:
36×t6\boxed{3^{6} \times t^{6}}

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