Problem 25 - Algebra 2

Question

Simplify. $$ \sqrt{49 m^{6}} $$

Step-by-Step Solution

Answer

The simplified expression is \(7m^3\).

Step 1: Simplify the Square Root of a Perfect Square

Identify that 49 is a perfect square. Its square root is 7, because \(7 \times 7 = 49\). Write this as: \( \sqrt{49} = 7 \).

Step 2: Simplify the Exponent Using Square Root

To simplify \( \sqrt{m^6} \), use the property \( \sqrt{a^2} = a \). Write \( m^6 \) as \((m^3)^2\), so \( \sqrt{m^6} = m^3 \).

Step 3: Combine the Results

Combine the results of steps 1 and 2 to write the simplified expression: \( 7m^3 \).

Key Concepts

Perfect Squares

A perfect square is a number that can be expressed as the square of an integer. For instance, 49 is a perfect square because it can be written as the product of 7 multiplied by itself: \( 7 \times 7 = 49 \). This means that the square root of 49 is an integer, which is 7.
Perfect squares are especially useful when simplifying square roots. They help in quickly identifying when a square root can be simplified to a whole number. Here's a quick way to spot a perfect square: if a number's square root results in an integer, then it's a perfect square.
Recognizing perfect squares help students simplify expressions efficiently and also strengthens number sense — knowing the squares of numbers up to at least 15 can be immensely helpful!

Exponent Rules

Exponent rules are guidelines that help to simplify expressions with powers of numbers or variables. They are key in handling terms that involve exponents when simplifying mathematical expressions.
Here are a few basic exponent rules to remember:

• Product of Powers: \( a^m \times a^n = a^{m+n} \)
• Power of a Power: \( (a^m)^n = a^{m \times n} \)
• Power of a Product: \( (ab)^n = a^n \times b^n \)

In the context of our exercise, we used these rules to simplify \( m^6 \). By recognizing that \( m^6 \) is \( (m^3)^2 \), we can take the square root as equal to \( m^3 \), using the power of a power rule. The more you actively use these rules, the more intuitive they become!

Properties of Square Roots

Square roots have specific properties that are vital for simplification processes. These properties make dealing with roots straightforward and predictable.
Some of the core properties include:

• Product Property: \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \)
• Quotient Property: \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \)
• Square Root of a Square: \( \sqrt{a^2} = |a| \) (absolute value ensures a positive result)

When simplifying our expression \( \sqrt{49m^6} \), we see that understanding these properties allows us to break down the problem. Recognizing that taking the square root of a square (like \( (m^3)^2 \)) simplifies directly to a single term, \( m^3 \), is crucial. Mastering these properties provides a strong foundation for tackling more complex expressions involving square roots.