Which Of These Is Equal To $5^0 + 2^1$?A. $6^0 - 6^1$B. $ 2 0 + 2 1 2^0 + 2^1 2 0 + 2 1 [/tex]C. $6^0 - 3^0$D. $2^0 + 1^1$

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Introduction

Exponents are a fundamental concept in mathematics, and understanding how to simplify them is crucial for solving various mathematical problems. In this article, we will explore the concept of exponents, learn how to simplify them, and apply this knowledge to solve a specific problem.

What are Exponents?

Exponents are a shorthand way of representing repeated multiplication. For example, $2^3$ means $2 \times 2 \times 2$, which equals $8$. Exponents are also known as powers or indices.

Simplifying Exponents

To simplify exponents, we need to understand the rules of exponentiation. Here are some key rules:

  • Product of Powers Rule: When multiplying two numbers with the same base, we add their exponents. For example, $2^3 \times 2^4 = 2^{3+4} = 2^7$.
  • Power of a Power Rule: When raising a power to another power, we multiply the exponents. For example, $(23)4 = 2^{3 \times 4} = 2^{12}$.
  • Zero Exponent Rule: Any number raised to the power of zero is equal to 1. For example, $2^0 = 1$.
  • Negative Exponent Rule: A negative exponent is equal to the reciprocal of the positive exponent. For example, $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$.

Simplifying the Given Expression

Now that we have a good understanding of exponents, let's simplify the given expression: $5^0 + 2^1$.

Using the zero exponent rule, we know that $5^0 = 1$. Therefore, the expression becomes:

1+211 + 2^1

Using the power of a power rule, we know that $2^1 = 2$. Therefore, the expression becomes:

1+21 + 2

Evaluating the expression, we get:

33

Comparing the Options

Now that we have simplified the given expression, let's compare it with the options:

A. $6^0 - 6^1$

Using the zero exponent rule, we know that $6^0 = 1$. Therefore, the expression becomes:

1−611 - 6^1

Using the power of a power rule, we know that $6^1 = 6$. Therefore, the expression becomes:

1−61 - 6

Evaluating the expression, we get:

−5-5

This is not equal to $3$, so option A is incorrect.

B. $2^0 + 2^1$

Using the zero exponent rule, we know that $2^0 = 1$. Therefore, the expression becomes:

1+211 + 2^1

Using the power of a power rule, we know that $2^1 = 2$. Therefore, the expression becomes:

1+21 + 2

Evaluating the expression, we get:

33

This is equal to the simplified expression, so option B is correct.

C. $6^0 - 3^0$

Using the zero exponent rule, we know that $6^0 = 1$ and $3^0 = 1$. Therefore, the expression becomes:

1−11 - 1

Evaluating the expression, we get:

00

This is not equal to $3$, so option C is incorrect.

D. $2^0 + 1^1$

Using the zero exponent rule, we know that $2^0 = 1$. Therefore, the expression becomes:

1+111 + 1^1

Using the power of a power rule, we know that $1^1 = 1$. Therefore, the expression becomes:

1+11 + 1

Evaluating the expression, we get:

22

This is not equal to $3$, so option D is incorrect.

Conclusion

In this article, we learned how to simplify exponents using various rules. We then applied this knowledge to simplify the given expression: $5^0 + 2^1$. We compared the simplified expression with the options and found that option B is correct.

Final Answer

Introduction

In our previous article, we explored the concept of exponents and learned how to simplify them using various rules. In this article, we will answer some frequently asked questions about simplifying exponents.

Q&A

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

A: A power and an exponent are often used interchangeably, but technically, an exponent is the number that is being raised to a power. For example, in the expression $2^3$, $2$ is the base and $3$ is the exponent.

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

A: To simplify an expression with multiple exponents, you need to follow the order of operations (PEMDAS):

  1. Evaluate any expressions inside parentheses.
  2. Evaluate any exponential expressions (e.g., $2^3$).
  3. Evaluate any multiplication and division expressions from left to right.
  4. Evaluate any addition and subtraction expressions from left to right.

For example, to simplify the expression $2^3 \times 2^4$, you would first evaluate the exponential expressions:

23=82^3 = 8

24=162^4 = 16

Then, you would multiply the results:

8×16=1288 \times 16 = 128

Q: What is the rule for simplifying negative exponents?

A: A negative exponent is equal to the reciprocal of the positive exponent. For example, $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$.

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

A: Any number raised to the power of zero is equal to 1. For example, $2^0 = 1$.

Q: Can I simplify an expression with a fractional exponent?

A: Yes, you can simplify an expression with a fractional exponent. For example, $2^{\frac{1}{2}} = \sqrt{2}$.

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

A: To simplify an expression with a negative base and a positive exponent, you need to follow the rule for negative exponents:

(−a)n=an(-a)^n = a^n

For example, $(-2)^3 = -8$.

Q: Can I simplify an expression with a variable base and a variable exponent?

A: Yes, you can simplify an expression with a variable base and a variable exponent. For example, $x^y$.

Q: How do I simplify an expression with multiple variables and exponents?

A: To simplify an expression with multiple variables and exponents, you need to follow the order of operations (PEMDAS) and apply the rules for exponents.

For example, to simplify the expression $x^2 \times y^3$, you would first evaluate the exponential expressions:

x2=x×xx^2 = x \times x

y3=y×y×yy^3 = y \times y \times y

Then, you would multiply the results:

(x×x)×(y×y×y)=x2×y3(x \times x) \times (y \times y \times y) = x^2 \times y^3

Conclusion

In this article, we answered some frequently asked questions about simplifying exponents. We hope that this article has helped you to better understand the concept of exponents and how to simplify them.

Final Tips

  • Always follow the order of operations (PEMDAS) when simplifying expressions with exponents.
  • Use the rules for exponents to simplify expressions with multiple exponents.
  • Be careful when simplifying expressions with negative exponents or fractional exponents.
  • Practice, practice, practice! The more you practice simplifying expressions with exponents, the more comfortable you will become with the rules and concepts.