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Chapter 7: Problem 9

Simplify each of the following as completely as possible. $$2^{3} 3^{4}$$

Short Answer

Expert verified
Write the original expression: Before simplifying, understand the original format of the product of powers Starting Expression: }}

Step by step solution

01

Title - Write the Original Expression

Start by writing down the original expression: The original expression is: }}

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Key Concepts

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

headline of the respective core concept
Understanding the original expression is the first crucial step. In our example, the expression we need to simplify is \(2^{3} 3^{4}\). This expression consists of two separate bases, 2 and 3, each raised to its own power鈥�3 and 4, respectively.

It's important to recognize that the operation called for is multiplication, as indicated by the adjacent placement of the terms and the absence of any other operators.
headline of the respective core concept
Exponents are a way to express repeated multiplication of a number by itself. For instance, \(2^{3}\) means multiplying 2 by itself three times:

\(2 \times 2 \times 2 = 8\).

Similarly, \(3^{4}\) represents multiplying 3 by itself four times:

\(3 \times 3 \times 3 \times 3 = 81\).

Understanding this allows us to convert exponents into their simplest form, which will be crucial for further simplification.
headline of the respective core concept
Simplification involves reducing the expression to its simplest form, using both multiplication and properties of exponents.

In our example, first compute each part separately:

\(2^{3} = 8\)
\(3^{4} = 81\)

Next, multiply the results of these computations:
8 \times 81 = 648

Thus, the simplified form of the original expression \(2^{3} 3^{4}\) is 648.

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Most popular questions from this chapter

Convert each number into standard notation. $$6.81 \times 10^{-2}$$

A certain sum of money is invested at \(10 \%,\) and twice that amount is invested at \(12 \% .\) If the annual interest from the two investments is \(\$ 408,\) how much is invested at each rate?

Multiply out \((x+6)(x+4)\) and \((x-6)(x-4) .\) What is the effect of switching both \(+\) signs to \(-\) signs?

Estimate the answer without actually carrying out the computation and make the most appropriate choice. If you multiply \(1.36 \times 10^{8}\) by \(1.18 \times 10^{-4},\) the result is closest to (a) 100 (b) 1000 (c) \(10,000\) (d) 0.001 (e) 0.0001

Suppose that the price \(p\) of a given item is related to the number of items sold \(x\) by the equation \(p=50-0.004 x\). The revenue \(R\) is computed by multiplying the price per item \(p\) by the number of items \(x\). Thus, \(R=x(50-0.004 x)\). Find the increase in revenue if \(x\) is increased by 500 .
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