Which Of These Is Equal To $5^0 + 2^1$?A. $2^0 + 1^1$B. \$6^0 - 3^0$[/tex\]C. $6^0 - 6^1$D. $2^0 + 2^1$
Introduction
Exponential expressions 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 exponential expressions, learn how to simplify them, and apply this knowledge to solve a specific problem.
What are Exponential Expressions?
Exponential expressions are mathematical expressions that involve the use of exponents, which are shorthand notations for repeated multiplication. An exponent is a small number that is placed above and to the right of a base number, indicating how many times the base number should be multiplied by itself. For example, the expression $2^3$ represents the number 2 multiplied by itself three times, or $2 \times 2 \times 2 = 8$.
Simplifying Exponential Expressions
Simplifying exponential expressions involves applying the rules of exponents to reduce the expression to its simplest form. There are several rules of exponents that we need to know in order to simplify exponential expressions:
- Product of Powers Rule: When multiplying two exponential expressions with the same base, we add the exponents. For example, $2^3 \times 2^4 = 2^{3+4} = 2^7$.
- Power of a Power Rule: When raising an exponential expression to a power, we multiply the exponents. For example, $(23)4 = 2^{3 \times 4} = 2^{12}$.
- Quotient of Powers Rule: When dividing two exponential expressions with the same base, we subtract the exponents. For example, $\frac{25}{23} = 2^{5-3} = 2^2$.
Simplifying the Given Expression
Now that we have learned how to simplify exponential expressions, let's apply this knowledge to simplify the given expression: $5^0 + 2^1$. Using the rules of exponents, we can simplify this expression as follows:
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5^0 = 1$, since any non-zero number raised to the power of 0 is equal to 1.
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2^1 = 2$, since any non-zero number raised to the power of 1 is equal to itself.
Therefore, the simplified expression is: $1 + 2 = 3$.
Comparing the Simplified Expression with the Options
Now that we have simplified the given expression, let's compare it with the options:
- A. $2^0 + 1^1$: This expression is equal to $1 + 1 = 2$, which is not equal to the simplified expression.
- B. $6^0 - 3^0$: This expression is equal to $1 - 1 = 0$, which is not equal to the simplified expression.
- C. $6^0 - 6^1$: This expression is equal to $1 - 6 = -5$, which is not equal to the simplified expression.
- D. $2^0 + 2^1$: This expression is equal to $1 + 2 = 3$, which is equal to the simplified expression.
Therefore, the correct answer is D. $2^0 + 2^1$.
Conclusion
Q: What is the difference between an exponent and a power?
A: An exponent is a small number that is placed above and to the right of a base number, indicating how many times the base number should be multiplied by itself. A power, on the other hand, is the result of raising a number to a certain exponent. For example, the expression $2^3$ represents the number 2 multiplied by itself three times, or $2 \times 2 \times 2 = 8$. The exponent is 3, and the power is 8.
Q: How do I simplify an exponential expression with a negative exponent?
A: To simplify an exponential expression with a negative exponent, we can use the rule that $a^{-n} = \frac{1}{a^n}$. For example, $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$. This rule allows us to rewrite negative exponents as fractions.
Q: Can I simplify an exponential expression with a zero exponent?
A: Yes, we can simplify an exponential expression with a zero exponent. Any non-zero number raised to the power of 0 is equal to 1. For example, $2^0 = 1$, since any non-zero number raised to the power of 0 is equal to 1.
Q: How do I simplify an exponential expression with a fractional exponent?
A: To simplify an exponential expression with a fractional exponent, we can use the rule that $a^{m/n} = \sqrt[n]{a^m}$. For example, $2^{3/4} = \sqrt[4]{2^3} = \sqrt[4]{8} = 2$. This rule allows us to rewrite fractional exponents as roots.
Q: Can I simplify an exponential expression with a negative base?
A: Yes, we can simplify an exponential expression with a negative base. For example, $(-2)^3 = -2 \times -2 \times -2 = -8$. When simplifying an exponential expression with a negative base, we can use the rule that $(-a)^n = a^n$ if n is even, and $(-a)^n = -a^n$ if n is odd.
Q: How do I simplify an exponential expression with multiple bases?
A: To simplify an exponential expression with multiple bases, we can use the rule that $a^m \times b^n = a^m \times b^n$. For example, $2^3 \times 3^4 = 2^3 \times 3^4$. This rule allows us to simplify exponential expressions with multiple bases by separating the bases and exponents.
Q: Can I simplify an exponential expression with a variable base?
A: Yes, we can simplify an exponential expression with a variable base. For example, $x^2$ is an exponential expression with a variable base. When simplifying an exponential expression with a variable base, we can use the rule that $a^m \times a^n = a^{m+n}$. For example, $x^2 \times x^3 = x^{2+3} = x^5$.
Conclusion
In conclusion, simplifying exponential expressions involves applying the rules of exponents to reduce the expression to its simplest form. By understanding these rules and applying them to various types of exponential expressions, we can simplify them and solve mathematical problems with ease.