Problem 117 - Elementary Algebra

Question

For the following problems, evaluate each numerical expression. $$ \frac{4^{-1}}{5^{-2}} $$

Step-by-Step Solution

Answer

Question: Evaluate the expression $$\frac{4^{-1}}{5^{-2}}$$. Answer: The value of the expression $$\frac{4^{-1}}{5^{-2}}$$ is 1.

Step 1: Rewrite the expression with positive exponents

To rewrite the expression with positive exponents, remember that for any nonzero number a and a negative exponent n, we have $$a^{-n} = \frac{1}{a^n}$$ Use this rule to rewrite the given expression: $$ \frac{4^{-1}}{5^{-2}} = \frac{\frac{1}{4^1}}{\frac{1}{5^2}} $$

Step 2: Simplify the expression

Now, we'll simplify the expression by multiplying the numerator and the denominator by the least common multiple of the denominatures, 20, and then compute the new value of the expression: $$ \frac{\frac{1}{4}}{\frac{1}{25}}\cdot\frac{20}{20} = \frac{1\cdot20}{4\cdot5} =\frac{20}{20} = 1 $$ Thus, the value of the expression $$\frac{4^{-1}}{5^{-2}}$$ is 1.

Key Concepts

Numerical Expressions

Numerical expressions are a key foundation in mathematics, comprised of numbers and operations like addition, subtraction, multiplication, and division. When presented with a numerical expression, the goal is often to perform calculations that lead to a single numerical result. For example, the numerical expression \( \frac{4^{-1}}{5^{-2}} \) involves the challenge of dealing with negative exponents. In such expressions, operations must be carried out in a specific order, often starting with simplifying exponents before moving on to other operations. Breaking down each part systematically aids in avoiding errors and reaching an accurate result.

Simplifying Expressions

Simplifying expressions means making them more straightforward and easier to understand without changing their value. When expressions involve negative exponents, like in \( \frac{4^{-1}}{5^{-2}} \), you can simplify by converting negative exponents into positive ones. Here’s how it works: For any nonzero number \( a \) and a negative exponent \( n \), you use the rule \( a^{-n} = \frac{1}{a^n} \). Apply this rule to transform each part of the expression into its positive form.

• \( 4^{-1} \) becomes \( \frac{1}{4^1} \)
• \( 5^{-2} \) becomes \( \frac{1}{5^2} \)

Now the expression \( \frac{4^{-1}}{5^{-2}} \) changes to \( \frac{\frac{1}{4}}{\frac{1}{25}} \). Once this transformation is complete, you can easily carry out additional operations, such as multiplication or factoring, to simplify further.

Positive Exponents

Positive exponents denote the number of times a base number is multiplied by itself. It simplifies many calculations and helps in evaluating expressions quickly. For instance, in the process to evaluate \( \frac{4^{-1}}{5^{-2}} \), converting negative exponents to positive exponents was a crucial step.
Once you rewrite the expression with positive exponents, \( 4^{-1} \) becomes \( \frac{1}{4^1} \), which is simply \( \frac{1}{4} \). Similarly, \( 5^{-2} \) becomes \( \frac{1}{25} \). Positive exponents reveal the true magnitude by showing how many times the base is used in the operation.

• Understanding positive exponents aids in rationalizing expressions.
• It helps to simplify expressions involving both numerical and variable elements.
• Using positive exponents in calculations ensures ease and accuracy in reaching the solution.

Expressing parts of mathematical calculations in this form highlights relationships and solves complex problems efficiently.