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Chapter 11: Problem 37

Use a calculator to evaluate each expression. $$\left(8 a^{3} b^{6}\right)^{1 / 3}$$

Short Answer

Expert verified
The simplified expression is \( 2ab^2 \).

Step by step solution

01

Identify the given expression

We are given the expression \( (8a^3b^6)^{1/3} \), which we need to simplify using the properties of exponents.
02

Apply the Power of a Power property

According to the Power of a Power property, \[(x^m)^n = x^{m imes n}.\]We can apply this to each part of the expression: \[(8a^3b^6)^{1/3} = 8^{1/3} imes (a^3)^{1/3} imes (b^6)^{1/3}.\]
03

Simplify each component separately

First, simplify each component:- \( 8^{1/3} \) is the cube root of 8, which is 2.- \( (a^3)^{1/3} = a^{3 imes rac{1}{3}} = a^{1} = a \).- \( (b^6)^{1/3} = b^{6 imes rac{1}{3}} = b^{2} \).Together these give us: \[ 2ab^2. \]
04

Combine the simplified expressions

Combine the simplified components from the previous step: \[ 2 imes a imes b^2 = 2ab^2. \]This is the simplified form of the given expression.

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Key Concepts

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

Power of a Power Property
The Power of a Power property is a useful tool in simplifying expressions that have exponents. Essentially, when you have a power raised to another power, you multiply the exponents. For instance, the property states that \[ (x^m)^n = x^{m \cdot n}. \]So, if you have \( (a^3)^{1/3} \),you multiply 3 by \( 1/3 \), resulting in \( a^1 \), which simplifies to \( a \).

Using this property helps you break down more complex exponential expressions into manageable parts. It can transform a long expression into something simpler. Remember the basic rule: multiply the exponents when you have a power of a power.
Simplifying Expressions
Simplifying expressions often involves reducing them to their simplest form. This process can include combining like terms, using properties of exponents, and performing arithmetic calculations.

In an expression like \[ (8a^3b^6)^{1/3}, \]you begin by separating the components:
  • \( 8^{1/3} \)
  • \( (a^3)^{1/3} \)
  • \( (b^6)^{1/3} \)
Each part is simplified individually:
  • \( 8^{1/3} \) simplifies to 2, since the cube root of 8 is 2.
  • \( (a^3)^{1/3} \) simplifies to \( a \), using the Power of a Power property.
  • \( (b^6)^{1/3} \) simplifies to \( b^2 \), also using the Power of a Power property.
Once simplified, these components are multiplied back together to get the final expression.
Simplification aims to make the expression as concise and straightforward as possible by applying foundational properties of algebra.
Cube Root Evaluation
Evaluating cube roots is straightforward once familiar with the cube root concept. A cube root is a number that, when multiplied by itself twice, gives the original number. For instance, the cube root of 8 is 2, because \( 2 \times 2 \times 2 = 8 \).

In expressions such as \( (8a^3b^6)^{1/3} \),the cube root is applied individually to each factor. This means:
  • For 8, \( 8^{1/3} = 2 \); thinking of what number cubed results in 8 helps find this quickly.
  • For \( a^3 \), the cube root \( (a^3)^{1/3} \) results in \( a \).
  • For \( b^6 \), \( (b^6)^{1/3} \) results in \( b^2 \) since \( b^2 \times b^2 \times b^2 = b^6 \).

Understanding cube roots simplifies expressions, particularly when some terms within are powers themselves. Knowing cube roots helps widely in algebra and number theory.

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Most popular questions from this chapter

Perform the indicated operations, expressing answers in simplest form with rationalized denominators. $$\sqrt[3]{5-\sqrt{17}} \sqrt[3]{5+\sqrt{17}}$$

$$\text {solve the given problems.}$$ By substitution, show that \(x=1-\sqrt{2}\) is a solution of the equation \(x^{2}-2 x-1=0\)

$$\text {Solve the given problems.}$$ In analyzing a tuned amplifier circuit, the expression \(\frac{2 Q}{\sqrt{\sqrt{2}-1}}\) is used. Rationalize the denominator.

Perform the indicated operations. A factor used in determining the performance of a solar-energy storage system is \((A / S)^{-1 / 4},\) where \(A\) is the actual storage capacity and \(S\) is a standard storage capacity. If this factor is \(0.5,\) explain how to find the ratio \(A / S\)

Solve the given problems. Express \(1 / 81(\text { a })\) as a power of 9 and \((b)\) as a power of 3.
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