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Chapter 10: Problem 61

Multiply. $$ y^{1 / 2}\left(y^{1 / 2}-y^{2 / 3}\right) $$

Short Answer

Expert verified
The expression simplifies to \( y - y^{7/6} \).

Step by step solution

01

Distribute the Multiplication

To multiply the expression \( y^{1/2}(y^{1/2} - y^{2/3}) \), begin by distributing \( y^{1/2} \) across each term in the parenthesis. This gives you two separate multiplications to perform: \( y^{1/2} imes y^{1/2} \) and \( y^{1/2} imes (-y^{2/3}) \).
02

Multiply the First Pair of Terms

In the multiplication \( y^{1/2} imes y^{1/2} \), use the property of exponents that states \( a^m imes a^n = a^{m+n} \). Therefore, \( y^{1/2} imes y^{1/2} = y^{1/2 + 1/2} = y^1 \). So the result of this multiplication is \( y \).
03

Multiply the Second Pair of Terms

Next, multiply \( y^{1/2} imes (-y^{2/3}) \). Again, use the property of exponents: \( y^{1/2} imes y^{2/3} = y^{1/2 + 2/3} \). To add the exponents, first convert them to have a common denominator. \( 1/2 = 3/6 \) and \( 2/3 = 4/6 \). Therefore, \( 1/2 + 2/3 = 3/6 + 4/6 = 7/6 \). Thus, the result of this multiplication is \(-y^{7/6} \).
04

Combine the Results

Now, combine the results of the two products. The expression after multiplication is \( y - y^{7/6} \). This is written as \( y - y^{7/6} \).

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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 principle in mathematics that allows us to simplify multiplication involving addition or subtraction within parentheses.
It states that:
  • Multiplying a single term by each term within a parenthesis can be expressed as multiplying the term with each term inside and then adding or subtracting the results.
In this exercise, we are using the distributive property to expand the expression: \( y^{1/2} (y^{1/2} - y^{2/3}) \).
Here鈥檚 how it works:
  • Distribute \( y^{1/2} \) to \( y^{1/2} \).
  • Distribute \( y^{1/2} \) to \(-y^{2/3} \).
This results in two separate terms that need to be multiplied individually.
Properties of Exponents
The properties of exponents make working with exponents straightforward and predictable. Let's dive into two essential rules:
  • Product of Powers: When multiplying terms with the same base, you add the exponents. This is shown as: \( a^m \times a^n = a^{m+n} \).
  • Negative Exponents: A negative exponent indicates the reciprocal of that base raised to the positive of the exponent. Though not involved in this problem, it's a good property to remember.
For this exercise, while multiplying \( y^{1/2} \times y^{1/2} \), we add the exponents:
  • \( 1/2 + 1/2 = 1 \), resulting in \( y^1 \).
While with \( y^{1/2} \times y^{2/3} \), we first align the fractions to a common denominator (e.g., convert \( 1/2 \) into \( 3/6 \) and \( 2/3 \) into \( 4/6 \)) and then add:
  • \( 3/6 + 4/6 = 7/6 \), giving us \( y^{7/6} \).
Fractional Exponents
Fractional exponents can seem intimidating at first, but they're quite manageable once you understand them.
  • A fractional exponent like \( y^{1/2} \) denotes a root, specifically a square root for exponents of \( 1/2 \). So, \( y^{1/2} = \sqrt{y} \).
  • More generally, \( y^{a/b} \) represents the \( b \)-th root of \( y^a \), which can also be expressed as \((\sqrt[b]{y})^a \).
In the given expression, you deal with fractional exponents such as \( y^{1/2} \) and \( y^{2/3} \).
These represent roots:
  • \( y^{1/2} \) is the square root,
  • \( y^{2/3} \) suggests the cube root of \( y \) squared.
By becoming familiar with fractional exponents, you unlock the ability to handle these complex expressions with ease.

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Most popular questions from this chapter

Write each integer as a product of two integers such that one of the factors is a perfect square. For example, write 18 as \(9.2,\) because 9 is a perfect square. $$ 75 $$

Use the properties of exponents to simplify each expression. Write with positive exponents. $$ \frac{\left(2 x^{1 / 5}\right)^{4}}{x^{3 / 10}} $$

Use rational expressions to write as a single radical expression. $$ \frac{\sqrt[3]{a^{2}}}{\sqrt[6]{a}} $$

Write each integer as a product of two integers such that one of the factors is a perfect square. For example, write 18 as \(9.2,\) because 9 is a perfect square. $$ 48 $$

Use rational exponents to simplify each radical. Assume that all variables represent positive numbers. $$ \sqrt[10]{a^{5} b^{5}} $$
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