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Chapter 10: Problem 16

Simplify. $$\frac{y^{7}}{y}$$

Short Answer

Expert verified
The simplified form is \(y^6\).

Step by step solution

01

Identify the Exponents

Note the exponents in the numerator and the denominator. In the expression \(\frac{y^{7}}{y}\), the exponent of the numerator is 7, and since the denominator can be rewritten as \(\frac{y^{7}}{y^1}\), the exponent of the denominator is 1.
02

Apply the Quotient Rule

Use the quotient rule for exponents, which states \(\frac{a^m}{a^n} = a^{m-n}\). Apply this rule to get \(\frac{y^{7}}{y^1} = y^{7-1}\).
03

Simplify the Exponent

Subtract the exponents: \(7 - 1 = 6\). Therefore, the expression simplifies to \(y^6\).

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Key Concepts

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

Exponents
Exponents are a shorthand way to express repeated multiplication of the same number by itself.
When you see something like \(y^7\), it means \(y\) multiplied by itself 7 times: \(y \cdot y \cdot y \cdot y \cdot y \cdot y \cdot y\).
Exponents make it easier to write and work with very large or very small numbers. They can also help simplify algebraic expressions.
Quotient Rule
The quotient rule for exponents allows us to simplify expressions where the same base variable is divided.
The rule states: \(\frac{a^m}{a^n} = a^{m-n}\).
This means when you divide two expressions with the same base, you can subtract the exponent of the denominator from the exponent of the numerator.
For example, in \(\frac{y^7}{y}\), the base is \(y\), the exponent of the numerator is 7, and the exponent of the denominator is 1. Applying the quotient rule gives us \(y^{7-1} = y^6\).
Simplifying Expressions
Simplifying expressions is all about making an expression easier to work with.
When dealing with exponents and the quotient rule, simplifying means to reduce the expression to the simplest form possible, using the laws of exponents.
In our example, \(\frac{y^7}{y}\) was simplified to \(y^6\) in three steps: identify the exponents, apply the quotient rule, and subtract the exponents.
Prealgebra
Prealgebra is the foundation of all advanced mathematical concepts.
It includes topics like basic operations, fractions, decimals, and the first principles of algebra, such as working with exponents.
Understanding how to simplify expressions that involve exponents is a crucial skill in prealgebra, paving the way for more complex topics in algebra and beyond.
Mastering prealgebra concepts ensures a strong mathematical foundation that will support future learning.

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Most popular questions from this chapter

Simplify the expression. $$\left(\frac{10}{y}\right)^{2}$$

Simplify the expression. $$\left(y^{2}\right)^{3}\left(y^{4}\right)^{2}$$

Fill in the missing polynomial. $$(\quad)(w-7)=w^{2}-4 w-21$$

Simplify. Write the answers with positive exponents only. $$5^{4} 5^{-7}$$

Write the opposite of the polynomial. $$2.4 m-5.1 n$$
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