Evaluate (1/(3h^5))^-4
Explanation of the Question:
This question is asking for the evaluation of an algebraic expression. Specifically, the expression provided is (1/(3h^5)) raised to the power of -4. Evaluating the expression involves performing exponentiation, which includes working with negative exponents and applying the rules of exponents to simplify the expression. The final answer should be a simplified form of the original expression, often expressed in terms of h without a negative exponent.
$\left(\left(\right. \frac{1}{3 h^{5}} \left.\right)\right)^{- 4}$
Answer
Solution:
Step 1:
Invert the base and change the negative exponent to positive: $(3h^5)^4$
Step 2:
Separate the constant and variable, applying the power rule: $3^4(h^5)^4$
Step 3:
Calculate the power of 3: $81(h^5)^4$
Step 4:
Handle the exponent on the variable:
Step 4.1:
Utilize the power of a power rule, $(a^m)^n = a^{mn}$: $81h^{5 \cdot 4}$
Step 4.2:
Complete the exponent multiplication: $81h^{20}$
Solution:"Evaluate the given expression by first converting the negative exponent to a positive one by taking the reciprocal of the base. Then, apply the power rule to both the constant and the variable separately. Calculate the power of the constant and then use the power of a power rule to simplify the variable's exponent, resulting in the final answer."
Knowledge Notes:
Negative Exponent Rule: $a^{-n} = \frac{1}{a^n}$. This rule states that to convert a negative exponent, you take the reciprocal of the base and make the exponent positive.
Power Rule: When raising a power to a power, you multiply the exponents. This is expressed as $(a^m)^n = a^{mn}$.
Product Rule for Exponents: When multiplying like bases, you add the exponents: $a^m \cdot a^n = a^{m+n}$.
Constants and variables are treated separately when they are raised to a power, and each is evaluated according to the exponent rules.
LaTeX is used to render mathematical expressions in a readable format, using commands that start with a backslash $\$and are enclosed in dollar signs $$$for inline expressions or double dollar signs $$$$for displayed equations.
