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Chapter 0: Problem 65

Simplify each expression. \(\sqrt{w^{12}}\)

Short Answer

Expert verified
The simplified expression is \({w^6}\).

Step by step solution

01

Identify the Given Expression

The given expression is \(\root{w^{12}}\). The goal is to simplify this expression.
02

Apply the Property of Square Roots

Recall that \(\root{a^2} = a\). This property can be extended to higher powers such that \(\root{a^n} = a^{n/2}\).
03

Simplify the Exponent

Using the property, rewrite the exponent: \(\root{w^{12}} = w^{12/2} = w^6\).

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Key Concepts

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

Properties of Exponents
Understanding the properties of exponents is crucial in algebra.
Exponents help us to express repeated multiplication compactly.
Here are some key properties of exponents that are useful in solving various algebraic expressions:
  • Product of Powers: When multiplying two expressions with the same base, add the exponents: \(a^m \times a^n = a^{m+n}\).
  • Power of a Power: When raising an exponent to another power, multiply the exponents: \( (a^m)^n = a^{m \times n} \).
  • Power of a Product: When raising a product to an exponent, raise each factor to the exponent: \( (ab)^m = a^m \times b^m \).
  • Quotient of Powers: When dividing two expressions with the same base, subtract the exponents: \(a^m / a^n = a^{m-n}\).
These properties make manipulating exponents easier and are essential when simplifying expressions.
Square Root Simplification
Simplifying square roots involves expressing the square root in its simplest form.
The square root of a number is a value that, when multiplied by itself, gives the original number.
For example: \( \sqrt{9} = 3 \) because \( 3 \times 3 = 9 \).
A useful property to know is \( \sqrt{a^2} = a \).
This property extends to higher powers as well, so we can generalize it: \(\sqrt{a^n} = a^{n/2}\).
Let's apply this property to an example:
  • Given: \( \sqrt{w^{12}} \).
  • Rewrite the exponent inside the square root: \( \sqrt{w^{12}} = w^{12/2} \).
  • Simplify the exponent: \( w^{12/2} = w^6 \).
Thus, \( \sqrt{w^{12}} = w^6 \). Simplifying square roots using exponent properties can make complex expressions more manageable.
Algebraic Expressions
Algebraic expressions consist of numbers, variables, and arithmetic operations.
They can be simplified by applying various algebraic rules.
Here鈥檚 an overview of key concepts:
  • Variables: Symbols (e.g., \( x, y, w \)) representing unknown values.
  • Constants: Fixed values like \( 2, 7, -3 \).
  • Coefficients: Numbers in front of variables indicating their multiplication, such as in \( 3x \) where \( 3 \) is the coefficient of \( x \).
  • Terms: The individual parts of an expression separated by plus or minus signs, e.g., in \( 3x + 4 \), \( 3x \) and \( 4 \) are terms.
To simplify algebraic expressions, combine like terms (terms with the same variables) and apply the order of operations.
When dealing with exponents and square roots, use the properties discussed earlier to make the expression simpler.
For example:
  • Simplify: \( 2x + 3x - x \).
  • Combine like terms: \( (2+3-1)x = 4x \).
Simplified expressions are easier to work with and essential for solving algebraic problems.

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Most popular questions from this chapter

Write the expanded form for \((a+b)(a-b)\).

Perform the indicated operations and simplify. $$ \left(3 w^{2}-7 z\right)\left(3 w^{2}+7 z\right) $$

Explain why \(x+\sqrt{7}\) is a polynomial, but \(\sqrt{x}+7\) is not a polynomial.

Multiply and simplify. Assume that all variable expressions represent positive real numbers. $$ 5 \sqrt{7}(3 \sqrt{7}-2 \sqrt{5}+3) $$

Multiply and simplify. $$ (w+7)(6 w-3) $$
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