What Does $4^2 \cdot 4^{-4}$ Equal?A. 4B. 1 4 \frac{1}{4} 4 1 C. 16D. 1 16 \frac{1}{16} 16 1
What does $4^2 \cdot 4^{-4}$ equal?
Understanding Exponents and Negative Exponents
In mathematics, exponents are a shorthand way of representing repeated multiplication. For example, $4^2$ means 4 multiplied by itself 2 times, which equals 16. However, when we encounter negative exponents, it can be a bit more challenging to understand. A negative exponent is essentially a fraction with the base as the numerator and 1 as the denominator. In other words, $4^{-4}$ is equivalent to $\frac{1}{4^4}$.
Evaluating the Expression
Now, let's evaluate the given expression $4^2 \cdot 4^{-4}$. To do this, we need to follow the order of operations (PEMDAS), which states that we should perform operations in the following order: Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).
In this case, we have two exponents, $4^2$ and $4^{-4}$. We can simplify these expressions separately before multiplying them together.
Simplifying the Exponents
4^2$ equals 16, as mentioned earlier. Now, let's simplify $4^{-4}$. As mentioned earlier, a negative exponent is equivalent to a fraction with the base as the numerator and 1 as the denominator. Therefore, $4^{-4}$ is equivalent to $\frac{1}{4^4}$.
To evaluate $4^4$, we need to multiply 4 by itself 4 times, which equals 256. Therefore, $4^{-4}$ equals $\frac{1}{256}$. Multiplying the Exponents Now that we have simplified the exponents, we can multiply them together. $4^2 \cdot 4^{-4}$ equals 16 multiplied by $\frac{1}{256}$. To multiply a number by a fraction, we can multiply the numerator of the fraction by the number and then divide by the denominator. In this case, we can multiply 16 by 1 (the numerator of the fraction) and then divide by 256 (the denominator of the fraction). Evaluating the Final Expression 16 multiplied by 1 equals 16. Now, we need to divide 16 by 256. To do this, we can divide both the numerator and the denominator by their greatest common divisor, which is 16. Dividing 16 by 16 equals 1, and dividing 256 by 16 equals 16. Therefore, $4^2 \cdot 4^{-4}$ equals $\frac{1}{16}$. Conclusion In conclusion, $4^2 \cdot 4^{-4}$ equals $\frac{1}{16}$. This is because we can simplify the exponents separately before multiplying them together, and then evaluate the final expression by multiplying the numerator and denominator by the same number. Key Takeaways Final Answer The final answer is D. .<br/>
Q&A: Understanding Exponents and Negative Exponents Frequently Asked Questions In the previous article, we discussed how to evaluate the expression $4^2 \cdot 4^{-4}$. However, we know that there are many more questions and concerns that readers may have. In this article, we will address some of the most frequently asked questions about exponents and negative exponents. Q: What is an exponent? A: An exponent is a shorthand way of representing repeated multiplication. For example, $4^2$ means 4 multiplied by itself 2 times, which equals 16. Q: What is a negative exponent? A: A negative exponent is equivalent to a fraction with the base as the numerator and 1 as the denominator. For example, $4^{-4}$ is equivalent to $\frac{1}{4^4}$. Q: How do I evaluate an expression with exponents? A: To evaluate an expression with exponents, you need to follow the order of operations (PEMDAS). This means that you should perform operations in the following order: Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). Q: Can I simplify exponents separately before multiplying them together? A: Yes, you can simplify exponents separately before multiplying them together. This can make it easier to evaluate the expression and avoid mistakes. Q: How do I multiply a number by a fraction? A: To multiply a number by a fraction, you can multiply the numerator of the fraction by the number and then divide by the denominator. Q: What is the difference between $4^2$ and $4^{-2}$? A: $4^2$ equals 16, while $4^{-2}$ equals $\frac{1}{16}$. This is because a negative exponent is equivalent to a fraction with the base as the numerator and 1 as the denominator. Q: Can I use exponents to simplify fractions? A: Yes, you can use exponents to simplify fractions. For example, $\frac{1}{4^2}$ is equivalent to $\frac{1}{16}$. 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 (PEMDAS). This means that you should perform operations in the following order: Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). Q: Can I use exponents to solve equations? A: Yes, you can use exponents to solve equations. For example, if you have the equation $x^2 = 16$, you can solve for x by taking the square root of both sides. Conclusion In conclusion, exponents and negative exponents are an important part of mathematics. By understanding how to evaluate expressions with exponents and negative exponents, you can simplify complex expressions and solve equations. We hope that this article has helped to answer some of the most frequently asked questions about exponents and negative exponents. Key Takeaways Final Answer The final answer is that exponents and negative exponents are a powerful tool in mathematics that can help you simplify complex expressions and solve equations. By understanding how to evaluate expressions with exponents and negative exponents, you can become a more confident and proficient mathematician.