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Chapter 1: Problem 15

Multiply and simplify. Leave the answer in exponential notation. $$ 5 x^{4} \cdot 4 x^{3} $$

Short Answer

Expert verified
20x^{7}

Step by step solution

01

Coefficient Multiplication

Multiply the coefficients (the numerical parts) of the terms. Here, you have 5 and 4. So, calculate: 5 * 4 = 20
02

Add Exponents

When multiplying terms with the same base, add the exponents. The base here is x, and the exponents are 4 and 3. So, calculate: x^{4+3} = x^{7}
03

Combine Results

Now combine the results from the previous steps: Coefficient result: 20 Exponential result: x^{7} So the final expression is: 20x^{7}

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Key Concepts

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

Exponential Notation
Exponential notation is a way to express repeated multiplication of the same number. For example, instead of writing \( x \times x \times x \times x \times x \times x \times x \), we can write \( x^7 \). The small number to the upper right of the base number is called the exponent. It tells you how many times to multiply the base number by itself. This makes it easier to write and understand large numbers or repeated multiplications.
Coefficient Multiplication
When multiplying expressions, it's important to handle the numerical parts, known as coefficients, separately from the variable parts with exponents. For the problem \( 5 x^4 \times 4 x^3 \):
  • The coefficients are 5 and 4.
  • Multiply these coefficients together: 5 * 4 = 20.
This multiplication involves basic arithmetic but is a crucial step to simplify the expression.
Adding Exponents
When multiplying terms that have the same base, like in the expression \( 5 x^4 \times 4 x^3 \), you need to add their exponents. Here鈥檚 how to do it:
  • The exponents in the problem are 4 and 3.
  • Because the base (which is 'x') is the same, you add the exponents: 4 + 3.
  • So, \( x^4 \times x^3 = x^{4+3} = x^7 \).
This step simplifies our expression to have only one exponent part associated with the same base.

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Most popular questions from this chapter

Use roster notation to write each set. The set of all odd natural numbers

Calculate using the rules for order of operations. If an expression is undefined, state this. $$ \frac{5 \cdot 2-4^{2}}{27-2^{4}} $$

Classify each statement as either true or false. The following sets are used: \(\mathrm{N}=\) the set of natural numbers; \(W=\) the set of whole numbers; \(\mathbb{Z}=\) the set of integers; \(\mathbb{C}=\) the set of rational numbers; \(\mathrm{H}=\) the set of irrational numbers; \(\mathrm{R}=\) the set of real numbers. $$ 4.3 \notin \mathbb{Z} $$

Write an equivalent expression using a commutative law. Answers may vary. $$ -9(a b) $$

Calculate using the rules for order of operations. If an expression is undefined, state this. $$ \frac{7 \cdot 3-5^{2}}{9+4 \cdot 2} $$
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