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Chapter 10: Problem 43

Simplify. See Examples 3 and 4 $$ \sqrt[4]{a^{8} b^{7}} $$

Short Answer

Expert verified
The simplified expression is \( a^2 b \sqrt[4]{b^3} \).

Step by step solution

01

Identify the Parts of the Expression

The expression under consideration is the fourth root of the product of powers: \( \sqrt[4]{a^{8} b^{7}} \). This consists of two parts under the root: \( a^8 \) and \( b^7 \).
02

Apply the Root to Each Part Separately

To simplify the expression, apply the fourth root separately to each component: \( \sqrt[4]{a^{8}} \) and \( \sqrt[4]{b^{7}} \).
03

Simplify \( \sqrt[4]{a^{8}} \)

Since 8 is a multiple of 4, simplify: \( \sqrt[4]{a^{8}} = a^{8/4} = a^2 \). Thus, it simplifies to \( a^2 \).
04

Simplify \( \sqrt[4]{b^{7}} \)

Since 7 is not a multiple of 4, break it into parts: \( \sqrt[4]{b^{7}} = \sqrt[4]{b^{4} \cdot b^{3}} = b^{4/4} \cdot \sqrt[4]{b^3} = b \cdot \sqrt[4]{b^{3}} \).
05

Combine the Simplified Parts

Now, combine the simplified parts: \( a^2 \cdot b \cdot \sqrt[4]{b^{3}} \). The expression \( \sqrt[4]{b^3} \) cannot be simplified further since 3 does not factor the root.

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Key Concepts

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

Exponents
Exponents are a way of expressing repeated multiplication of a number by itself. When you see a number or variable with a power, like \( a^8 \), it means that \( a \) is multiplied by itself 8 times. So, \( a^8 = a \times a \times a \times a \times a \times a \times a \times a \).
Understanding exponents is crucial when working with algebraic expressions, especially when simplifying radicals or roots. Here's what you need to know about exponents:
  • If the exponent is 1, it means the value itself: \( x^1 = x \).
  • If the exponent is 0, any value except 0 raised to the power of zero is 1: \( x^0 = 1 \).
  • When multiplying with like bases, add the exponents: \( x^a \times x^b = x^{a+b} \).
  • When dividing with like bases, subtract the exponents: \( x^a / x^b = x^{a-b} \).
  • When raising a power to another power, multiply the exponents: \((x^a)^b = x^{a\times b} \).
In the given problem, we're applying such rules to effectively break down and simplify expressions with roots.
Roots
Roots are the inverse operation of exponents. When you take a root, like the square root \( \sqrt{x} \) or fourth root \( \sqrt[4]{x} \), you're looking for a number that, when raised to the given power, equals \( x \). Roots help in simplifying higher power expressions.
Here's what to keep in mind about roots:
  • The square root of a number \( y \) is a number \( x \) such that \( x^2 = y \).
  • The fourth root \( \sqrt[4]{y} \) finds a number \( x \) such that \( x^4 = y \).
  • Taking a root is equivalent to raising a number to a fractional exponent: \( \sqrt[4]{y} = y^{1/4} \).
In simplification processes, roots work hand in hand with exponents. For instance, in \( \sqrt[4]{a^8} \), the root simplifies by dividing the exponent by the root number: \( a^{8/4} = a^2 \). This concept highlights the ease roots bring to otherwise complex expressions.
Algebraic Expressions
Algebraic expressions are mathematical phrases that can include numbers, variables, and operations. In this context, simplifying algebraic expressions involving radicals and exponents is a key skill.
Some quick tips about algebraic expressions:
  • Combine like terms, which are terms with the same variables to the same power.
  • Use the order of operations (PEMDAS/BODMAS) to simplify expressions correctly.
  • Apply the properties of exponents and roots as discussed above when handling complex expressions.
In the exercise \( \sqrt[4]{a^{8} b^{7}} \), you face variables \( a \) and \( b \) raised to exponents within a radical. Using algebraic properties such as breaking down exponents and applying roots individually is essential to simplify expressions accurately. As in \( a^2 \cdot b \cdot \sqrt[4]{b^{3}} \), combining simplified elements forms a neat, simpler expression.

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Most popular questions from this chapter

Which of the following are not real numbers? Explain why \(\sqrt[3]{-64}\) is a real number.

Use rational exponents to simplify each radical. Assume that all variables represent positive numbers. $$ \sqrt[9]{y^{6} z^{3}} $$

Simplify each exponential expression. $$\left(-14 a^{5} b c^{2}\right)\left(2 a b c^{4}\right)$$

Use a calculator to write a four-decimal-place approximation of each number. $$ 20^{1 / 5} $$

For Exercises 105 through \(108,\) do not use a calculator. \(\sqrt{1000}\) is closest to a. 10 b. 30 c. 100 d. 500
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