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Chapter 7: Problem 100

Perform the indicated operation and simplify. Assume that all variables represent positive real numbers. Write the answer using radical notation. $$ \sqrt[4]{a^{2} b}(\sqrt[3]{a^{2} b}-\sqrt[5]{a^{2} b^{2}}) $$

Short Answer

Expert verified
{a^{1/2}}{b^{1/3}}-{a^{2/5}}{b^{2/5}}.

Step by step solution

01

Distribute the Expression

Distribute \(\sqrt[4]{a^{2} b}\) to both terms inside the parenthesis: \(\sqrt[4]{a^{2} b} \cdot \sqrt[3]{a^{2} b} - \sqrt[4]{a^{2} b} \cdot \sqrt[5]{a^{2} b^{2}}\).
02

Combine Radicals with Same Base

Combine the radicals of the same base: \(\sqrt[4]{a^{2} b} \cdot \sqrt[3]{a^{2} b} = \sqrt[4 \times 3]{(a^{2} b)^{3} (a^{2} b)^{4}} = \sqrt[12]{a^{6} b^{4}}\) and \(\sqrt[4]{a^{2} b} \cdot \sqrt[5]{a^{2} b^{2}} = \sqrt[4 \times 5]{(a^{2} b)^{5} (a^{2} b^{2})^{4}} = \sqrt[20]{a^{8} b^{8}}\).
03

Simplify Each Radical

Simplify \(\sqrt[12]{a^{6} b^{4}} = \sqrt[12]{(a^{6}) (b^{4})} = a^{6/12} b^{4/12} = a^{1/2} b^{1/3}\). Similarly, simplify \(\sqrt[20]{a^{8} b^{8}} = \sqrt[20]{a^{8} b^{8}} = a^{8/20} b^{8/20} = a^{2/5} b^{2/5}\).
04

Substitute Back Simplified Radicals

Substitute the simplified radicals back into the expression: \(a^{1/2} b^{1/3} - a^{2/5} b^{2/5}\).

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Key Concepts

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

Distributive Property
The distributive property is a fundamental algebraic rule that allows us to multiply a term across terms inside parentheses. In this exercise, we need to distribute \[ \text{鈭歔4]{a^{2} b}} \] to each term within the parenthesis. This means we multiply \[ \text{鈭歔4]{a^{2} b}} \] first by \[ \text{鈭歔3]{a^{2} b}} \], and then by \[ \text{鈭歔5]{a^{2} b^2}} \]. This step breaks down the problem and makes it easier to handle.
Combining Radicals
Combining radicals with the same base involves using the properties of exponents to simplify them into a single radical. For instance, \[ \text{鈭歔4]{a^{2} b} \cdot \text{鈭歔3]{a^{2} b}} \] can be combined as \[ \text{鈭歔12]{(a^{2} b)^{3} (a^{2} b)^{4}}} \]. Here, 12 is the least common multiple of 4 and 3, which simplifies the complex multiplication of radicals. Similarly, for \[ \text{鈭歔4]{a^{2} b} \cdot \text{鈭歔5]{a^{2} b^{2}}}} \], we simplify the product into a single radical expression \[ \text{鈭歔20]{(a^2 b)^{5} (a^{2} b^{2})^{4}}} \].
Simplifying Radicals
After combining radicals, the next step involves simplifying them by converting the expressions inside the radicals into fractional exponents. For instance, simplifying \[ \text{鈭歔12]{a^{6} b^{4}}} = a^{6/12} b^{4/12} \], reduces to \[ a^{1/2} b^{1/3} \]. Simplifying another radical like \[ \text{鈭歔20]{a^{8} b^{8}}} \] follows similarly: it converts to \[ a^{8/20} b^{8/20} \], and then reduces further to \[ a^{2/5} b^{2/5} \]. Simplifying radicals also makes it easier to read and solve expressions.
Fractional Exponents
Fractional exponents are another way to represent radicals. For example, \[ a^{1/2} \] is the same as \[ \text{鈭歛} \] and \[ b^{1/3} \] is the same as \[ \text{鈭沚} \]. This makes it easier to represent and simplify radicals. Fractional exponents convert radical notation into a more straightforward format which simplifies the handling of algebraic operations. In our exercise, we used fractional exponents to simplify expressions like \[ \text{鈭歔12]{a^{6} b^{4}}} \] into \[ a^{1/2} b^{1/3} \] and \[ \text{鈭歔20]{a^{8} b^{8}}} \] into \[ a^{2/5} b^{2/5} \]. This step concludes the simplification of the original expression and makes it easy to arrive at the final answers.

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Most popular questions from this chapter

Find the midpoint of the segment with the given endpoints. $$ (9,2 \sqrt{3}) \text { and }(-4,5 \sqrt{3}) $$

Perform the indicated operation and simplify. Assume that all variables represent positive real numbers. Write the answer using radical notation. $$ \sqrt{b^{3}} \sqrt[5]{b^{4}} $$

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Simplify. $$\left(\frac{1}{2}-\frac{1}{3} i\right)^{2}-\left(\frac{1}{2}+\frac{1}{3} i\right)^{2}$$
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