Problem 39 - College Algebra

Question

Simplify each expression. $$m^{\frac{5}{2}} \sqrt{289}$$

Step-by-Step Solution

Answer

\(17m^{\frac{5}{2}}\)

Step 1: Simplify the Square Root

First, simplify the square root: \(\sqrt{289}\). The square root of 289 is 17 because \(17 \times 17 = 289\). So, we can rewrite the expression as \(m^{\frac{5}{2}} \times 17\).

Step 2: Combine the Terms

Now, we combine the simplified square root and the exponentiation: \(17m^{\frac{5}{2}}\). The expression is now a product of a coefficient and a power of \(m\).

Step 3: Final Expression

The expression is now in its simplest form: \(17m^{\frac{5}{2}}\). No further simplification is possible.

Key Concepts

Exponentiation

Exponentiation is a mathematical operation involving two numbers, the base and the exponent. When you see something like \(m^{\frac{5}{2}}\), it means you are multiplying the base \(m\) by itself a certain number of times. The exponent \(\frac{5}{2}\) is a fractional exponent, which can be confusing at first. Here's how to handle it:

• The numerator (5) tells you how many times to multiply the base by itself.
• The denominator (2) indicates that you are also performing a square root operation.

This means \(m^{\frac{5}{2}}\) can be rewritten as \(\left(m^5\right)^{\frac{1}{2}}\), which is the square root of \(m^5\). Instead of calculating the powers and roots separately, understanding this relationship simplifies our calculations, especially when simplifying algebraic expressions.

Square Roots

Square roots are the inverse operation of squaring a number. When you square a number, you multiply it by itself. The square root, however, asks "what number, when multiplied by itself, gives this original number?"

In the exercise, we dealt with \(\sqrt{289}\). To simplify this, we look for a number that multiplies by itself to get 289. In this case, 17 multiplied by 17 equals 289, so \(\sqrt{289} = 17\).

Understanding square roots is crucial for simplifying expressions because it allows you to break down numbers into more manageable parts. This simplification is based on identifying pairs of identical factors, which is helpful in handling larger or more complicated numbers in algebraic expressions.

Algebraic Expressions

Algebraic expressions are combinations of numbers and variables like \(m\), along with operations such as addition, subtraction, multiplication, and division. In our exercise, we saw \(m^{\frac{5}{2}} \times 17\). This expression combines exponentiation, multiplication, and a numerical coefficient.

When simplifying an algebraic expression, follow these guidelines:

• Simplify inside any parentheses first if needed.
• Perform operations involving exponents and roots.
• Multiply or divide coefficients (numbers) and variables separately, then combine them into a single term.

In our example, the term \(m^{\frac{5}{2}}\) is kept distinct, while the \(\sqrt{289}\) was simplified to 17, allowing us to write the simpler product \(17m^{\frac{5}{2}}\). This understanding assists in evaluating and rewriting expressions into more usable or understandable forms, making problem-solving easier.