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Chapter 2: Problem 40

Simplify the given expression as completely as possible. $$y^{4} y^{6}$$

Short Answer

Expert verified
Simplified expression is \(y^{10}\).

Step by step solution

01

Identify the Expression

The given expression is \(y^{4} y^{6}\). This is a multiplication of two powers with the same base.
02

Apply the Laws of Exponents

According to the laws of exponents, when multiplying like bases, add the exponents. Therefore, \(y^{4} y^{6} = y^{4+6}\).
03

Simplify the Exponents

Add the exponents together: \(4 + 6 = 10\). Thus, \(y^{4+6} = y^{10}\).

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Key Concepts

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

Simplifying Expressions
Simplifying expressions is a fundamental skill in algebra which helps in breaking down complex problems into manageable parts. It involves reducing the expression to its simplest form without changing its value. Simplification can include combining like terms, factoring, or using algebraic rules such as the laws of exponents.

For example, in the expression provided in the exercise, we have two terms with the same base, which is 'y'. The goal is to simplify this expression as much as possible. By applying relevant algebraic rules, the expression becomes much easier to handle.
Multiplying Exponents
When you multiply exponents with the same base, you follow a specific rule from the laws of exponents: you add the exponents together. This rule is based on the distributive property of multiplication over addition.

Consider the example provided: \(y^{4} y^{6}\). Here, both exponents are attached to the same base, which is 'y'. According to the laws of exponents, when you multiply these, you simply add the exponents: \(y^{4+6}\), which results in \(y^{10}\).
This approach simplifies the multiplication process and avoids lengthy expansion of terms.
Algebra Basics
Understanding the basics of algebra is crucial for solving more complex problems later on. Algebra involves using symbols and letters to represent numbers and quantities in equations and expressions.

Some fundamental concepts in algebra include:
  • Variables - symbols that represent unknown numbers (like 'y' in the exercise)
  • Constants - fixed values
  • Coefficients - numbers that multiply variables
  • Operations - such as addition, subtraction, multiplication, and division

For instance, in the expression \(y^{4} y^{6}\), 'y' is a variable, and the exponents (4 and 6) are constants. Understanding how to manipulate these elements is essential for mastering algebra.
By mastering these basics, you'll gain the tools needed to approach a wide variety of mathematical problems with confidence.

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Most popular questions from this chapter

Simplify by combining like terms whenever possible. $$m+9 m-3 n-n-10 m$$

Evaluate the given expression for \(x=2, y=-3,\) and \(z=-4.\) $$\frac{z}{x}-y$$

Evaluate the given expression for \(x=0.24\) and \(y=-0.5 .\) Round off to the nearest thousandth where necessary. $$10 x^{3}+20 y$$

Evaluate the given expression for \(x=2, y=-3,\) and \(z=-4.\) $$y(x-z)^{2}$$

Simplify by combining like terms whenever possible. $$0.08 x+0.09(x+1,500)+0.12(4,500-3 x)$$
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