/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 3 Multiply each pair of numbers.\(... [FREE SOLUTION] | 91影视

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Chapter 7: Problem 3

Multiply each pair of numbers.\(6 \times 6^{15}\)

Short Answer

Expert verified
Answer: When multiplying 6 by \(6^{15}\), the result is \(6^{16}\).

Step by step solution

01

Identify the base and exponents

In this multiplication problem, the base is 6 and the exponents are 1 and 15.
02

Multiply using the exponent rule

According to the exponent rule, multiply the numbers by adding their exponents: $$6^1 \times 6^{15} = 6^{1+15}$$
03

Compute the sum of exponents

Add the exponents 1 and 15: $$1 + 15 = 16$$
04

Write the final result

Using the resulting exponent, the multiplication of the given pair of numbers is: $$6^1 \times 6^{15} = 6^{16}$$

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Key Concepts

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

Base and Exponent
The base and exponent form the foundational building blocks of exponentiation, a key concept in understanding how to simplify expressions involving powers. In the context of mathematical operations, the base is the number that is being raised to a power, and the exponent indicates how many times the base is multiplied by itself.

For example, in the expression \(6^{15}\), 6 is the base and 15 is the exponent. This means that we are multiplying 6 by itself 15 times. It's crucial to note that \(6^{1}\) is simply 6, since any number raised to the power of one is the number itself.
Multiplying Exponents
When it comes to multiplying exponents with the same base, there is a simple yet powerful rule that can help streamline the process. This rule states that when you multiply powers with the same base, you keep the base the same and add the exponents.

Consider the example \(6^{1} \times 6^{15}\). Despite appearing daunting, this is straightforward once you apply the exponent rule. Since both terms have the base of 6, you can add the exponents together to simplify the expression to \(6^{16}\).
Mathematical Operations
Mastering mathematical operations like addition, subtraction, multiplication, and division is fundamental in manipulating and solving expressions with exponents. Addition and subtraction are used when dealing with the exponents themselves, while multiplication and division rules apply when you're combining or reducing terms with exponents.

In our exercise, we specifically use multiplication. Knowing these operations and how they interact with the rules for exponents is crucial for solving problems in algebra and beyond.
Exponent Addition
When we talk about exponent addition, we are referring to adding the exponents of numbers with the same base during the process of multiplication. This operation is an integral part of simplifying expressions with multiple powers of the same base.

Always remember, to perform exponent addition, the bases must be identical. Our original problem \(6^{1} \times 6^{15}\) is a classic example where we add the exponents 1 and 15 to get the new exponent, 16, for the common base of 6. It is a simple addition problem: \(1 + 15 = 16\), leading to the final answer of \(6^{16}\).

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Most popular questions from this chapter

Raise each number to the power indicated.\(\left(y^{2}\right)^{5}\)

Multiply each pair of numbers.\(54^{2} \times 54^{9}\)

Multiply each pair of numbers.\(4^{-3} \times 4^{3}\)

Multiply each pair of numbers.\(2^{3} \times 2^{5}\)

v\(54^{2} \times 54^{9}\)
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