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Chapter 9: Problem 13

Multiply the following expressions. \(y^{3} \cdot y^{2} \cdot y\)

Short Answer

Expert verified
The expression simplifies to \( y^6 \).

Step by step solution

01

Understand the Problem

We need to multiply the expressions \( y^3 \), \( y^2 \), and \( y \). This involves combining the powers of \( y \) into a single term.
02

Apply the Product of Powers Rule

When multiplying like bases, we add the exponents according to the rule \( a^m imes a^n = a^{m+n} \). Here, we add the exponents of \( y \): \( 3 + 2 + 1 \).
03

Simplify the Expression

Adding the exponents from the previous step, we get: \( 3 + 2 + 1 = 6 \). Therefore, the simplified form of the expression is \( y^6 \).

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Key Concepts

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

Exponentiation
Exponentiation is a mathematical operation that involves expression of a number, called the base, being raised to the power of an exponent.
It essentially tells you how many times to multiply the base by itself.
For example, in the expression \( y^3 \), the base is \( y \) and the exponent is \( 3 \). This indicates:
  • \( y \) multiplied by itself, three times, i.e., \( y \times y \times y \).
Exponentiation is a powerful tool in algebra because it simplifies the representation of long multiplication chains. Knowing how to correctly interpret and manipulate exponents is crucial for solving more complex algebraic expressions.
Simplifying Expressions
In algebra, simplifying expressions is about reducing them into their simplest form. This involves combining like terms and applying mathematical rules efficiently.
In the given exercise, \( y^3 \cdot y^2 \cdot y \), each term involves \( y \) raised to different powers.
By applying the product of powers rule, we simplify the multiplication by adding the exponents. This simplifies the expression to \( y^6 \).
  • Each original exponent 鈥� 3, 2, 1 鈥� is summed, resulting in a single term.
  • The process transforms a complex multiplication into a simple power of a single base.
Knowing how to simplify expressions is key for solving equations and understanding how changing parts of an expression affects the whole.
Multiplying Like Bases
Multiplying like bases is one of the foundational skills in algebra that involves combining powers of the same base.
The Product of Powers Rule states that when we multiply two exponents with the same base, we simply add the exponents together.
For example, in the expression \( a^m \times a^n = a^{m+n} \), if the base \( a \) is the same, then the exponents can be added.
  • This rule drastically decreases the complexity of calculations.
  • Instead of multiple multiplication steps, you perform one addition step.
Applying this rule in the exercise means summarizing the powers of \( y \) 鈥� thus \( y^3 \cdot y^2 \cdot y \) becomes \( y^{3+2+1} = y^6 \). Familiarity with this process is vital in efficiently solving algebraic problems and is a stepping stone to more advanced mathematical operations.

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Most popular questions from this chapter

Simplify each expression, and write all answers in scientific notation. $$\frac{\left(3.5 \times 10^{-4}\right)\left(4.2 \times 10^{5}\right)}{7 \times 10^{3}}$$

Simplify each expression, and write all answers in scientific notation. $$\frac{\left(4 \times 10^{9}\right)\left(6 \times 10^{5}\right)}{8 \times 10^{3}}$$

Expand and multiply. $$(x-5)^{2}$$

Simplify the following expressions. \(\left(10 x^{5}\right)^{2}\)

Find the value of each of the following expressions when \(x=4\). $$-5 x+6$$
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