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Chapter 10: Problem 14

Multiply. $$d^{2} d^{3} d^{4}$$

Short Answer

Expert verified
d^9

Step by step solution

01

Understand the problem

The problem requires multiplying the expressions with the same base, which is the variable 'd'. To solve this, follow the laws of exponents.
02

Recall the laws of exponents

One of the fundamental laws of exponents states that when multiplying like bases, add the exponents: \[ d^a \times d^b = d^{a+b} \]
03

Apply the exponent rule

Apply this rule to the given expression: \[ d^2 \times d^3 \times d^4 \]. Add the exponents together: \[ 2 + 3 + 4 = 9 \]
04

Write the final answer

Using the sum of the exponents, the expression simplifies to: \[ d^9 \]

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Key Concepts

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

multiplication of exponents
When we talk about multiplying exponents, it means we are dealing with terms that have exponents and the same base. Here, our base is \(d\). The multiplication of exponents might seem tricky at first, but it follows a simple rule.
If we look at the expression from the exercise, \( d^2 \times d^3 \times d^4 \), we clearly see the repetition of the same base \(d\). When you multiply these, the multiplication rule of exponents tells us to add the exponents.
So instead of multiplying straight away, focus on the exponents when the bases are the same.
like bases
In the world of exponents, the term 'like bases' simply means that the terms you are dealing with share the same base. For instance, in our exercise, each part of \( d^2 \times d^3 \times d^4 \) has \(d\) as the base.
This concept is crucial because multiplying or dividing terms with different bases follows different rules.
Whenever you spot like bases being multiplied, remember it's your cue to add up the exponents, simplifying the problem into something easier to manage.
exponent addition
Exponent addition is a cornerstone of working with exponents. When multiplying terms with the same base, simply add their exponents. Here's how it works:
Consider the expression \( d^2 \times d^3 \times d^4 \). Each part has the same base \(d\). The rule tells us to add the exponents: 2, 3, and 4.
Add them up: \(2 + 3 + 4 = 9\). So the expression becomes \(d^9\).
This method saves time and avoids confusion. Anytime you have like bases being multiplied, sum the exponents for a quick solution.

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Most popular questions from this chapter

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