Question

How would you make 1/2 smaller?

1. x^-1/x^2=
2. (x^-2)^-1
3. x^-5*x^4
4. 1/x^-2*x^4
5. Compare 10^2 to 10^-2

Answer 1

[AI-generated answer — Bot GPT 3.5]

1. To make 1/2 smaller, you can multiply the numerator or divide the denominator by any number greater than 1. For example, multiplying the numerator by 0.5 would result in 0.5/2, which is smaller than 1/2.

2. (x^-2)^-1 can be simplified by applying the negative exponent rule, which states that when a base with a negative exponent is raised to a negative exponent, it becomes positive. Therefore, (x^-2)^-1 simplifies to x^2.

3. To simplify x^-5 * x^4, you can apply the exponent rule for multiplying variables with the same base. When multiplying variables with the same base, you add their exponents. Therefore, x^-5 * x^4 simplifies to x^(-5+4) = x^-1.

4. To simplify 1/x^-2 * x^4, you can first apply the negative exponent rule to x^-2, which gives you 1/x^2 * x^4. Then, you can apply the exponent rule for multiplying variables with the same base to x^2 * x^4, which results in x^(2+4) = x^6.

5. When comparing 10^2 to 10^-2, you can convert them to decimal form to determine which is smaller. 10^2 is equal to 100, and 10^-2 is equal to 0.01. Therefore, 10^-2 is smaller than 10^2.

Answer 2

1. what two operations can you apply exponent rules?

2. how are exponent rules different from operation rules with numbers?
3. Given the following list of expressions: -2^3, -2^-2, and (-3)^3
3 part a. Evaluate each expression.
3 part b. Order the expressions from least to greatest
4. (2^2)(2^8)=
5. (2x^3)^4
6. (7^2y^3z^4)^2
7. 3m(m/8)^4

Answer 3

[AI-generated answer — Bot GPT 3.5]

1. Two operations that can be applied using exponent rules are multiplication of numbers with the same base and different exponents, and raising a number with an exponent to another exponent.

2. Exponent rules deal specifically with manipulating exponents and their properties. They provide guidelines and shortcuts for simplifying and performing calculations involving exponents. On the other hand, operation rules with numbers (e.g., addition, subtraction, multiplication, division) focus on manipulating numbers and performing basic arithmetic operations.

3.
3a. Evaluating each expression:
-2^3 can be written as -(2^3) = -(2*2*2) = -8.
-2^-2 can be written as -1/(2^2) = -1/4.
(-3)^3 is equal to -3*(-3)*(-3) = -27.

3b. Ordering the expressions from least to greatest:
-2^-2, -2^3, (-3)^3

4. (2^2)(2^8) can be simplified using the exponent rule for multiplication of numbers with the same base. The rule states that when multiplying numbers with the same base, you add their exponents. Therefore, (2^2)(2^8) equals 2^(2+8) = 2^10.

5. (2x^3)^4 can be simplified by applying the exponent rule for raising a power to another exponent. The rule states that when raising a number with an exponent to another exponent, you multiply the exponents. Therefore, (2x^3)^4 equals 2^4 * (x^3)^4 = 16x^12.

6. (7^2y^3z^4)^2 can be simplified by applying the exponent rule for raising a power to another exponent. The rule states that when raising a power to another exponent, you multiply the exponents. Therefore, (7^2y^3z^4)^2 simplifies to 7^(2*2) * (y^3)^2 * (z^4)^2 = 49y^6z^8.

7. 3m(m/8)^4 can be simplified by applying the exponent rule for raising a power to another exponent. The rule states that when raising a power to another exponent, you multiply the exponents. Therefore, 3m(m/8)^4 becomes 3m(m^4/8^4) = 3m(m^4/4096).