Problem 39 - College Algebra

Question

For the following exercises, simplify each expression. $$ m^{\frac{5}{2}} \sqrt{289} $$

Step-by-Step Solution

Answer

The simplified expression is \( 17m^{\frac{5}{2}} \).

Step 1: Simplify the square root

First, identify the square root present in the expression and simplify it if possible. In this expression, we have \(\sqrt{289}\). Recognizing that 289 is a perfect square, note that \(289 = 17^2\). Therefore, \( \sqrt{289} = 17 \). Now the expression becomes \( m^{\frac{5}{2}} \times 17 \).

Step 2: Simplify the expression multiplication

Multiply the simplified result from Step 1 by the remaining term in the expression. The expression now is \( m^{\frac{5}{2}} \times 17 \). Write this multiplication as \( 17m^{\frac{5}{2}} \).

Step 3: Verify the expression

Double-check each simplification step to ensure there are no errors. Confirm that \( \sqrt{289} = 17 \) and that the multiplication is correctly noted, leading to the final simplified expression of \( 17m^{\frac{5}{2}} \).

Key Concepts

Square Roots

Square roots are an essential concept in mathematics, especially when simplifying expressions. A square root refers to a number which, when multiplied by itself, gives the original number. For instance, if you have the number 289, its square root is 17. This is because when you multiply 17 by itself (i.e., 17 \( \times \) 17), the product is 289.

• Understanding square roots is useful in breaking down complex expressions into simpler forms.
• Recognizing patterns such as perfect squares helps in rapid simplification.

In the expression \(\sqrt{289}\), knowing that 289 is a perfect square allows us to directly substitute \(\sqrt{289}\) with 17 in the expression \(m^{\frac{5}{2}} \sqrt{289}\), leading to a more straightforward calculation.

Perfect Squares

Perfect squares are numbers that arise from squaring whole numbers. They are vital insights in simplifying expressions involving square roots or quadratic terms. For example, 1, 4, 9, 16, up to 289 and beyond, are perfect squares as they can be expressed as the square of integers (like \(1^2 = 1\), \(4^2 = 16\), \(17^2 = 289\)).

• A perfect square simplifies the process of square rooting because it provides a direct integer result without decimals.
• Recognizing perfect squares speeds up the simplification steps in many mathematical expressions.

In the original problem \(\sqrt{289}\), noting that 289 is a perfect square allows us to simplify efficiently and accurately. We substitute \(\sqrt{289}\) with 17, streamlining the expression simplification process.

Exponentiation

Exponentiation is a shortcut for repeated multiplication of a number by itself. It is represented with an exponent, a small raised number next to the base number. For instance, in the expression \(m^{\frac{5}{2}}\), \(m\) is our base and \(\frac{5}{2}\) is the exponent. This means we are taking the square root of \(m^5\), simplifying it in terms of root and power.

• Exponents can be fractional, indicating both roots and powers in expressions.
• Knowing how to manipulate exponents is crucial when rearranging or simplifying expressions in algebra.

After simplifying \( \sqrt{289}\) to 17, the expression becomes \(17m^{\frac{5}{2}}\). The exponent tells us how \(m\) is to be treated, ensuring precision when multiplying or further manipulating the expression. Understanding and applying exponent rules are crucial for handling and simplifying mathematical operations effectively.