B. $\left(8 \times 10^7\right) + \left(4 \times 10^6\right) =$ $\square$ (Use The Multiplication Symbol In The Math Palette As Needed.)

Mar 1, 2025 by ADMIN 136 views

**Understanding Exponents and Multiplication in Mathematics**

Introduction

In mathematics, exponents and multiplication are two fundamental operations that are used to simplify complex calculations. Exponents are used to represent repeated multiplication of a number, while multiplication is used to find the product of two or more numbers. In this article, we will explore the concept of exponents and multiplication, and how they are used to solve mathematical problems.

What are Exponents?

Exponents are a shorthand way of representing repeated multiplication of a number. For example, the expression $2^3$ represents the product of 2 multiplied by itself 3 times, which is equal to $2 \times 2 \times 2 = 8$ . Exponents are written as a small number raised to a power, and are used to simplify complex calculations.

Understanding the Order of Operations

When working with exponents and multiplication, it is essential to follow the order of operations. The order of operations is a set of rules that dictate the order in which mathematical operations should be performed. The order of operations is as follows:

Parentheses: Evaluate expressions inside parentheses first.
Exponents: Evaluate any exponential expressions next.
Multiplication and Division: Evaluate any multiplication and division operations from left to right.
Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Solving the Problem

Now that we have a basic understanding of exponents and multiplication, let's solve the problem:

b. $\left(8 \times 10^7\right) + \left(4 \times 10^6\right) =$ $\square$

To solve this problem, we need to follow the order of operations. First, we need to evaluate the expressions inside the parentheses.

$\left(8 \times 10^7\right) = 8 \times 10,000,000 = 80,000,000$

$\left(4 \times 10^6\right) = 4 \times 1,000,000 = 4,000,000$

Now that we have evaluated the expressions inside the parentheses, we can add the two results together.

$80,000,000 + 4,000,000 = 84,000,000$

Therefore, the solution to the problem is:

$\left(8 \times 10^7\right) + \left(4 \times 10^6\right) = 84,000,000$

Conclusion

In conclusion, exponents and multiplication are two fundamental operations in mathematics that are used to simplify complex calculations. By following the order of operations and understanding the concept of exponents, we can solve mathematical problems with ease. In this article, we solved the problem $\left(8 \times 10^7\right) + \left(4 \times 10^6\right) =$ $\square$ and found the solution to be $84,000,000$ .

Real-World Applications

Exponents and multiplication have numerous real-world applications. For example, in finance, exponents are used to calculate compound interest, while multiplication is used to calculate the total value of an investment. In science, exponents are used to represent the magnitude of a quantity, while multiplication is used to calculate the product of two or more quantities.

Tips and Tricks

Here are some tips and tricks to help you master exponents and multiplication:

Use the multiplication symbol: When working with exponents, use the multiplication symbol to represent repeated multiplication.
Follow the order of operations: When working with exponents and multiplication, follow the order of operations to ensure that you are performing the operations in the correct order.
Use parentheses: Use parentheses to group expressions and ensure that the operations are performed in the correct order.
Practice, practice, practice: The more you practice working with exponents and multiplication, the more comfortable you will become with these operations.

Common Mistakes

Here are some common mistakes to avoid when working with exponents and multiplication:

Not following the order of operations: Failing to follow the order of operations can lead to incorrect results.
Not using parentheses: Failing to use parentheses can lead to confusion and incorrect results.
Not practicing: Failing to practice working with exponents and multiplication can lead to a lack of understanding and incorrect results.

Conclusion

Q: What is the difference between an exponent and a power?

A: An exponent is a small number that is raised to a power, while a power is the result of raising a number to an exponent. For example, in the expression $2^3$ , the 3 is the exponent and the 2 is the base.

Q: How do I evaluate an expression with multiple exponents?

A: To evaluate an expression with multiple exponents, you need to follow the order of operations. First, evaluate any exponential expressions inside parentheses, then evaluate any exponential expressions outside parentheses. Finally, evaluate any multiplication and division operations from left to right.

Q: What is the rule for multiplying exponents with the same base?

A: When multiplying exponents with the same base, you add the exponents. For example, in the expression $2^3 \times 2^4$ , the result is $2^{3+4} = 2^7$ .

Q: How do I evaluate an expression with a negative exponent?

A: To evaluate an expression with a negative exponent, you need to follow the rule that $a^{-n} = \frac{1}{a^n}$ . For example, in the expression $2^{-3}$ , the result is $\frac{1}{2^3} = \frac{1}{8}$ .

Q: What is the rule for dividing exponents with the same base?

A: When dividing exponents with the same base, you subtract the exponents. For example, in the expression $\frac{2^4}{2^3}$ , the result is $2^{4-3} = 2^1 = 2$ .

Q: How do I evaluate an expression with a zero exponent?

A: To evaluate an expression with a zero exponent, you need to follow the rule that $a^0 = 1$ . For example, in the expression $2^0$ , the result is 1.

Q: What is the rule for raising a power to a power?

A: When raising a power to a power, you multiply the exponents. For example, in the expression $(2^3)^4$ , the result is $2^{3 \times 4} = 2^{12}$ .

Q: How do I evaluate an expression with a fractional exponent?

A: To evaluate an expression with a fractional exponent, you need to follow the rule that $a^{m/n} = \sqrt[n]{a^m}$ . For example, in the expression $2^{3/4}$ , the result is $\sqrt[4]{2^3} = \sqrt[4]{8} = 2$ .

Q: What is the rule for evaluating an expression with a negative base and a positive exponent?

A: When evaluating an expression with a negative base and a positive exponent, you need to follow the rule that $(-a)^n = a^n$ if $n$ is even, and $(-a)^n = -a^n$ if $n$ is odd. For example, in the expression $(-2)^3$ , the result is $-2^3 = -8$ .

Q: How do I evaluate an expression with a negative base and a negative exponent?

A: To evaluate an expression with a negative base and a negative exponent, you need to follow the rule that $(-a)^{-n} = \frac{1}{a^n}$ . For example, in the expression $(-2)^{-3}$ , the result is $\frac{1}{2^3} = \frac{1}{8}$ .

Conclusion

In conclusion, exponents and multiplication are two fundamental operations in mathematics that are used to simplify complex calculations. By following the order of operations and understanding the rules for exponents, we can solve mathematical problems with ease. In this article, we answered some common questions about exponents and multiplication, and provided examples to illustrate the rules.