Problem 102 Simplify each expression. $$10... [FREE SOLUTION]

Chapter 7: Problem 102

Simplify each expression. $$10 \sqrt{m}-\sqrt{16 m}$$

Short Answer

Expert verified

The simplified expression is $6 \sqrt{m}$.

Step by step solution

Simplify the Radical

First, simplify $\text{ }\sqrt{16m} $.Since 16 is a perfect square, it can be simplified: \[ \sqrt{16m} = \sqrt{16} \cdot \sqrt{m} = 4 \sqrt{m} \]

Substitute Simplified Radical

Replace $ \sqrt{16 m} $ in the original expression with its simplified form: \[10\sqrt{m} - 4 \sqrt{m} \]

Combine Like Terms

Combine like terms by subtracting the coefficients of $ \sqrt{m} $: \[ (10 - 4) \sqrt{m} = 6 \sqrt{m} \] Thus, the expression simplifies to \[6 \sqrt{m}\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Perfect Squares

In simplifying radical expressions, recognizing perfect squares is crucial. Perfect squares are numbers like 1, 4, 9, 16, 25, etc., which are products of an integer with itself. For example, 16 is a perfect square because it can be written as $4 \times 4$. This property helps us to simplify radicals.

For instance, when dealing with the term $ \sqrt{16m}$, we can separate it into the product of two square roots: \sqrt{16} \cdot \sqrt{m}\. Since \sqrt{16} = 4 \, the expression simplifies to \4 \sqrt{m}\.

Recognizing and simplifying perfect squares first can significantly simplify the overall expression, making further steps, such as combining like terms, much easier.

Like Terms

Like terms are terms in an expression that have the same variable raised to the same power. In our example, the terms 10\sqrt{m} and 4\sqrt{m} are considered like terms because they both contain the square root of the variable m.

Like terms can be added or subtracted from each other by combining their coefficients. For instance, in the expression $10\sqrt{m} - 4\sqrt{m}$, we can subtract the coefficients (10 and 4) while keeping the \sqrt{m} part the same. This simplifies our expression to \6\sqrt{m}\.

Combining like terms is a fundamental step in simplifying algebraic expressions, ensuring we get the simplest form of the expression.

Coefficients

Coefficients are the numerical factors placed in front of variables or radicals in an algebraic expression. In the expression $10\sqrt{m} - 4\sqrt{m}$, the numbers 10 and 4 are coefficients of \sqrt{m}\.

When simplifying expressions with radical terms, we focus on combining the coefficients of like terms. In our example, the like terms both have the \sqrt{m}\ part. The coefficients (10 and 4) can be subtracted from one another: $10 - 4$. This yields the new coefficient of 6 in the resulting term \6\sqrt{m}\.

Understanding coefficients helps in performing algebraic manipulations correctly, ensuring simplified and accurate results.

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91影视

Simplify each expression. $$10 \sqrt{m}-\sqrt{16 m}$$

Short Answer

Step by step solution

Simplify the Radical

Substitute Simplified Radical

Combine Like Terms

Key Concepts

Perfect Squares

Like Terms

Coefficients

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