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Chapter 1: Problem 35

Simplify. $$\left(3 x^{56}\right)\left(8 x^{23}\right)$$

Short Answer

Expert verified
The simplified expression is \(24x^{79}\).

Step by step solution

01

Distribute the Coefficients

First, we will multiply the coefficients of the terms. Take the coefficient of the first term, 3, and multiply it by the coefficient of the second term, 8. This will give us:\[ 3 \times 8 = 24 \]
02

Combine the Exponents

Next, we need to combine the powers of the variable \(x\). When multiplying terms with the same base, we add the exponents. Here, we add the exponents 56 and 23:\[ 56 + 23 = 79 \]
03

Combine Results

Finally, we combine the results from Step 1 and Step 2. The simplified form of the expression \( (3x^{56})(8x^{23}) \) is:\[ 24x^{79} \]

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Key Concepts

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

Exponents
Exponents are a way to express repeated multiplication of the same number or variable. For example, in the expression \(x^{56}\), 56 is the exponent and tells you how many times you need to multiply \(x\) by itself. Exponents allow us to write long multiplication sequences in a compact manner.

When multiplying terms with the same base, like \(x\) in our example, you add the exponents together. This is because multiplication accumulates the powers. For instance, multiplying \(x^{56}\) by \(x^{23}\) involves adding the exponents: \(56 + 23 = 79\). This is a fundamental property of exponents and helps simplify expressions efficiently.
  • If the bases are the same, you just add the exponents: \(a^m \times a^n = a^{m+n}\).
  • When in doubt, think of exponents as shorthand for multiplication.
Coefficients
Coefficients are the numerical factors in terms of a polynomial expression. In the expression \(3x^{56}\), 3 is the coefficient. It represents how many times the term \(x^{56}\) is repeated or scaled. Coefficients help us quantify expressions by giving each term its specific 'weight' or intensity.

When simplifying expressions, especially multiplication, you multiply the coefficients together separately from the variables. For example, in the original exercise, the coefficients 3 and 8 are multiplied together, resulting in 24. This step is crucial to get the numerical part of your simplified expression.
  • Multiply coefficients directly to find the new coefficient in simplified expressions.
  • Coefficients are always the numbers in front of the variables.
Simplifying Expressions
Simplifying expressions is about reducing them to the simplest or most compact form possible while retaining the same value. You achieve simplification by combining like terms and applying mathematical operations according to established rules.

In our exercise, simplifying means multiplying the polynomial terms \((3x^{56})\) and \((8x^{23})\).
  • First, multiply the coefficients: \(3 \times 8 = 24\).
  • Second, add the exponents of the same variable base: \(x^{56} \times x^{23} = x^{79}\).
  • Finally, combine these results to get the simplified expression: \(24x^{79}\).
Each step in simplification follows basic mathematical principles and ensures the expression is as straightforward and easy to understand as possible.

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Most popular questions from this chapter

Simplify the expression. $$(3 x+1)^{6}\left(\frac{1}{2}\right)(2 x-5)^{-1 / 2}(2)+(2 x-5)^{1 / 2}(6)(3 x+1)^{5}(3)$$

Simplify the expression. $$\frac{(3 x+2)^{1 / 2}\left(\frac{1}{3}\right)(2 x+3)^{-2 / 3}(2)-(2 x+3)^{1 / 3}\left(\frac{1}{2}\right)(3 x+2)^{-1 / 2}(3)}{\left[(3 x+2)^{1 / 2}\right]^{2}}$$

Evaluate the pair of expressions for \(x=1,2,3\) 4, and 5 by constructing a table of values. Discuss whether or not the two expressions might be equal. $$\frac{113 x^{3}+280 x^{2}-150 x}{22 x^{3}+77 x^{2}-100 x-350}, \frac{3 x}{2 x+7}+\frac{4 x^{2}}{1.1 x^{2}-5}$$

Simplify the expression, and rationalize the denominator when appropriate. $$\sqrt{\frac{1}{3 x^{3} y}}$$

Simplify the expression. $$\frac{4}{3 s+1}-\frac{11}{(3 s+1)^{2}}$$
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