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Chapter 4: Problem 21

Find each product or quotient. Express using exponents. $$(10 x)\left(4 x^{7}\right)$$

Short Answer

Expert verified
The product is \(40x^8\).

Step by step solution

01

Distribute the Product

First, distribute the multiplication across each term in the expression \((10x)(4x^7)\).We can rewrite this as \(10 \cdot 4 \cdot x \cdot x^7\).
02

Multiply the Constants

Multiply the constant numbers 10 and 4 together:\[10 \times 4 = 40\].
03

Multiply the Variables Using Exponents

Apply the property of exponents \(x^a \times x^b = x^{a+b}\). Here, \(x\) is equivalent to \(x^1\), so multiply \(x^1 \cdot x^7\):\[x^{1+7} = x^8\].
04

Combine the Results

Combine the results from multiplying the constants and the variables. The final product is:\[40x^8\].

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Key Concepts

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

Exponents
Exponents are a foundational concept in mathematics that simplify repeated multiplication of the same number or variable. If you see something like \(x^7\), it means that the number or variable \(x\) is being multiplied by itself 7 times. For beginners:
  • \(x^2\) means \(x \times x\)
  • \(x^3\) means \(x \times x \times x\)
  • \(x^4\) or any higher exponent follows the same pattern
Exponents serve as a shorthand, making expressions and calculations more concise. Understanding exponents is crucial when learning to simplify expressions, especially when dealing with algebraic formulas. This concept not only reduces the amount of writing but aids in easily grasping more complex mathematical ideas.
Multiplication of Variables
Multiplying variables, especially when they have exponents, involves a straightforward rule. For example, when you multiply \(x\) times \(x^7\), you are essentially adding the exponents. This is because:
  • The single \(x\) is really \(x^1\). Therefore, you compute \(x^1 \cdot x^7\) by adding the exponents: \(1 + 7\).
  • The result becomes \(x^8\), meaning \(x\) is used as a factor eight times.
Remember that you only add exponents when bases are the same. This concept is powerful when simplifying or expanding algebraic expressions, allowing you to maintain precision and clarity.
Properties of Exponents
When working with exponents, understanding their properties is key in simplifying mathematical expressions. The main properties to keep in mind include:
  • The product of powers property: When multiplying like bases, you add their exponents, as in \(x^a \cdot x^b = x^{a+b}\).
  • Power of a power property: If you have an expression \((x^a)^b\), you multiply the exponents to get \(x^{a\cdot b}\).
  • Power of a product property: To raise a product to a power, as in \((ab)^n\), you multiply each factor in the product: \(a^n \cdot b^n\).
These properties help streamline calculations in algebra and beyond, making seemingly complex problems simpler. They apply to a wide range of scenarios, from basic arithmetic to advanced calculus, showcasing the flexibility and efficiency that properties of exponents provide.
Distributive Property
The distributive property is a crucial arithmetic rule that allows you to multiply a single term across terms within a parenthesis, such as in \((a + b) \cdot c = ac + bc\). In the context of the problem \((10x)(4x^7)\):
  • Use the distributive property to rewrite the expression as \(10 \cdot 4 \cdot x \cdot x^7\).
  • This means each of the terms in the first parentheses is multiplied by each of the terms in the second one.
The distributive property effectively breaks down complicated expressions into more manageable parts. This makes calculations easier and ensures accuracy. Understanding how to apply this property is beneficial for simplifying expressions and solving equations in algebra.

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Most popular questions from this chapter

Find each product or quotient. Express using exponents. $$3 a^{2} \cdot 5 a^{2}$$

Find each product or quotient. Express using exponents. $$\frac{y^{9}}{y^{2}}$$

Keyboarding speed can be determined by using the formula \(s=\frac{w-10 r}{m}\) where s represents the speed of words typed per minute, \(w\) represents the number of words typed, \(r\) represents the number of errors, and \(m\) represents the total number of minutes typed. If Esteban received a keyboard speed of 80 words per minute and typed 530 words in 6 minutes, how many errors did he make?

Simplify each fraction. If the fraction is already in simplest form, write simplified. $$\frac{40 d}{42 d}$$

Numbers can also be expressed in expanded form. Example \(1: 13,548=10,000+3000+500+40+8\) \(=\left(1 \times 10^{4}\right)+\left(3 \times 10^{3}\right)+\left(5 \times 10^{2}\right)+\left(4 \times 10^{1}\right)+\left(8 \times 10^{0}\right)\) Example \(2: 0.568=0.5+0.06+0.008\) $$ =\left(5 \times 10^{-1}\right)+\left(6 \times 10^{-2}\right)+\left(8 \times 10^{-3}\right) $$ Write each number in expanded form. $$0.5875$$
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