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Chapter 3: Problem 7

Simplify the expression. a. \(\left(64 x^{9}\right)^{1 / 3}\) b. \(\left(25 x^{3} y^{4}\right)^{1 / 2}\)

Short Answer

Expert verified
The simplified expressions are: a. \(\left(64 x^{9}\right)^{1 / 3} = 4x^{3}\) b. \(\left(25 x^{3} y^{4}\right)^{1 / 2} = 5x^{\frac{3}{2}}y^2\)

Step by step solution

01

Apply the power to a power rule to each term.

We have two terms inside the brackets: \(64\) and \(x^{9}\). Apply the power to a power rule: \((64)^{1 / 3} \cdot (x^{9})^{1 / 3}\).
02

Simplify each term separately.

Now, we simplify each term: - \((64)^{1 / 3}\) is the cube root of \(64\), which equals \(4\). - \((x^{9})^{1 / 3}\) simplifies to \(x^{(9 \cdot \frac{1}{3})}\) = \(x^{3}\).
03

Combine the simplified terms for the answer.

Combine the simplified terms: \(4 \cdot x^{3}\). So, \(\left(64 x^{9}\right)^{1 / 3} = 4x^{3}\). b. Simplify \(\left(25 x^{3} y^{4}\right)^{1 / 2}\):
04

Apply the power to a power rule to each term.

We have three terms inside the brackets: \(25\), \(x^{3}\), and \(y^4\). Apply the power to a power rule: \((25)^{1 / 2} \cdot (x^{3})^{1 / 2} \cdot (y^{4})^{1 / 2}\).
05

Simplify each term separately.

Now, we simplify each term: - \((25)^{1 / 2}\) is the square root of \(25\), which equals \(5\). - \((x^{3})^{1 / 2}\) simplifies to \(x^{(3 \cdot \frac{1}{2})}\) = \(x^{\frac{3}{2}}\). - \((y^{4})^{1 / 2}\) simplifies to \(y^{(4 \cdot \frac{1}{2})}\) = \(y^{2}\).
06

Combine the simplified terms for the answer.

Combine the simplified terms: \(5 \cdot x^{\frac{3}{2}} \cdot y^{2}\). So, \(\left(25 x^{3} y^{4}\right)^{1 / 2} = 5x^{\frac{3}{2}}y^2\).

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Key Concepts

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

Power to a Power Rule
When you see an expression where a base with an exponent is raised to another power, like \((a^m)^n\), you apply the "power to a power" rule. This rule states that you multiply the exponents, resulting in \(a^{m \cdot n}\). This concept is crucial when simplifying exponential expressions, especially when dealing with nested exponents.
For example, consider \((x^9)^{\frac{1}{3}}\). According to the power to a power rule, multiply 9 and \(\frac{1}{3}\):
  • \(x^{9 \cdot \frac{1}{3}} = x^3\).
Applying this rule can significantly simplify complex expressions, making it a powerful tool in algebra.
Cube Root
The cube root of a number is a value that, when multiplied by itself three times, gives the original number. The cube root is represented by the radical symbol with a small 3 above it, like \(\sqrt[3]{a}\). Finding the cube root is common in algebra simplification problems.
When you encounter an expression like \(64^{\frac{1}{3}}\), you're essentially finding the cube root of 64. Which means:
  • \(\sqrt[3]{64} = 4\) because \(4 \times 4 \times 4 = 64\).
Understanding this concept helps break down and simplify exponential expressions that involve fractional exponents.
Square Root
Square roots are quite similar to cube roots but simpler since they deal with multiplying a number by itself just two times. The square root of a number \(a\) is written as \(\sqrt{a}\). Finding the square root is used commonly in solving equations and simplifying expressions.
Take for example, the expression \(25^{\frac{1}{2}}\). Here, you are tasked with finding the square root of 25:
  • \(\sqrt{25} = 5\) because \(5 \times 5 = 25\).
Mastering how to find square roots can make it much easier to simplify expressions involving exponents.
Exponents Simplification
Simplifying expressions with exponents involves a variety of operations including breaking down powers, multiplying bases, and utilizing both square and cube roots. When taking expression like \((x^3)^{\frac{1}{2}}\) or \((y^4)^{\frac{1}{2}}\), you simply apply the power to a power rule:
  • For \((x^3)^{\frac{1}{2}}\), it becomes \(x^{3 \cdot \frac{1}{2}} = x^{\frac{3}{2}}\).
  • For \((y^4)^{\frac{1}{2}}\), it simplifies to \(y^{4 \cdot \frac{1}{2}} = y^2\).
Each step in this simplification process is significant for reducing expressions to their simplest form and helps in solving complex algebraic expressions efficiently.

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Most popular questions from this chapter

According to data obtained from the CBO, the total federal debt (in trillions of dollars) from 2001 through 2006 is given by $$ f(t)=5.37 e^{0.07 s t} \quad(1 \leq t \leq 6) $$ where \(t\) is measured in years, with \(t=1\) corresponding to 2001\. What was the total federal debt in 2001 ? In 2006 ?

Use logarithms to solve the equation for \(t\). $$\frac{1}{3} e^{-3 t}=0.9$$

Use logarithms to solve the equation for \(t\). $$12-e^{0.4 t}=3$$

Use the laws of logarithms to solve the equation. $$\log x-\log (x+6)=-1$$

Express each equation in logarithmic form. $$5^{-3}=\frac{1}{125}$$
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