Question - Solving Exponential Expressions
mathsolving exponential expressionsexponents propertiespower of a power propertyquotient of powers property
Solution:
To solve the given expression, we will use the properties of exponents:The expression is $$ \left( \frac{(5^3)^2}{(5^2)^4} \right) $$.Now, let's apply the power of a power property for exponents, which states that $$(a^m)^n = a^{(m \cdot n)}$$:For the numerator, $$(5^3)^2 = 5^{(3 \cdot 2)} = 5^6$$,For the denominator, $$(5^2)^4 = 5^{(2 \cdot 4)} = 5^8$$,Now, rewrite the expression with the simplified exponents:$$ \frac{5^6}{5^8} $$.Using the quotient of powers property, which states that $$a^m/a^n = a^{(m-n)}$$, we subtract the exponents:$$ 5^{(6-8)} = 5^{-2} $$.The negative exponent rule states that $$a^{-n} = \frac{1}{a^n}$$:Therefore, $$5^{-2} = \frac{1}{5^2}$$.Since $$5^2 = 25$$, the expression simplifies to:$$ \frac{1}{25} $$.Therefore, the answer is option (1) $$ \frac{1}{25} $$.
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