Simplify (7^-2a)/(b^-1)
This problem involves simplification of an expression with negative exponents. Specifically, the expression is a fraction where the numerator contains 7 raised to the power of negative 2a, and the denominator contains b raised to the power of negative 1. To simplify this expression, you should apply the exponent rules, which dictate how to handle negative exponents and simplify fractions that contain them.
$\frac{7^{- 2} a}{b^{- 1}}$
Answer
Solution:
Step 1:
Apply the negative exponent rule $a^{-n} = \frac{1}{a^n}$ to rewrite $7^{-2a}$ in the denominator: $\frac{a}{\frac{1}{7^{2a}} \cdot b^{-1}}$.
Step 2:
Use the negative exponent rule again to bring $b^{-1}$ to the numerator: $\frac{ab}{7^{2a}}$.
Step 3:
Calculate the square of 7, which is $7^2 = 49$, to simplify the expression: $\frac{ab}{49}$.
Knowledge Notes:
The negative exponent rule is a fundamental concept in algebra which states that for any nonzero number $b$ and any integer $n$, the expression $b^{-n}$ is equivalent to $\frac{1}{b^n}$. This rule is used to simplify expressions with negative exponents by transforming them into fractions with positive exponents.
Another important concept is the power rule, which is used to simplify expressions where an exponent is raised to another power. For example, $(b^m)^n = b^{mn}$.
In the context of the given problem, these rules are applied to simplify the expression by moving terms with negative exponents from the numerator to the denominator or vice versa, and then simplifying the numerical exponent to obtain the final result.
