Use the Product Rule of Exponents to simplify
1. Use the Product Rule of Exponents to simplify 6^10 x 6^5 (put answer in exponent form)
2. Find the numerical equivalent of 8^9 x 8^-6
3. Find the value of x to make this exponential equation true…
5^x ÷ 5^5 = 5^12
4. Use the Power Rule of Exponents to simplify (9^3)^6—answer in exponent form.
5. Apply the Properties of Integer Exponents to generate equivalent expressions to
(6^-3)^7 x 6^20
━━━━━━━━ answer as an improper fraction. (this is a fraction problem..numerator above the line and denominator below it)
6^-2
1. Using the Product Rule of Exponents, we can simplify 6^10 x 6^5 as 6^(10+5) = 6^15.
2. To find the numerical equivalent of 8^9 x 8^-6, we can use the Product Rule of Exponents and simplify it as 8^(9+(-6)) = 8^3. Thus, the numerical equivalent is 512.
3. To find the value of x in the equation 5^x ÷ 5^5 = 5^12, we can use the Quotient Rule of Exponents and simplify it as 5^(x-5) = 5^12. Since the bases are the same, we can equate the exponents and solve for x. Therefore, x - 5 = 12, and solving for x gives us x = 17.
4. Using the Power Rule of Exponents, we can simplify (9^3)^6 as 9^(3*6) = 9^18.
5. To generate equivalent expressions for (6^-3)^7 x 6^20 / 6^-2, we can use the Product and Quotient Rules of Exponents. Simplifying, we have (6^(-3*7)) x 6^20 / 6^(-2) = 6^-21 x 6^20 / 6^-2. Applying the Quotient Rule, we get 6^(-21+20) / 6^-2 = 6^-1 / 6^-2. Using the Quotient Rule again, we have 6^-1 x 6^2 = 6^(2-1) = 6^1. Thus, the equivalent expression is 6^1, which simplifies to 6.
1. To simplify 6^10 x 6^5 using the Product Rule of Exponents, we add the exponents together. So we have 6^(10+5).
Simplified answer: 6^15
2. To find the numerical equivalent of 8^9 x 8^-6, we use the Product Rule of Exponents by adding the exponents together. So we have 8^(9+(-6)) = 8^3.
Simplified answer: 8^3 = 512
3. To find the value of x in the equation 5^x ÷ 5^5 = 5^12, we can use the Quotient Rule of Exponents. According to the rule, when dividing with the same base, we subtract the exponents. So we have x-5 = 12.
Solving for x, we add 5 to both sides of the equation:
x = 12 + 5
x = 17
The value of x that makes the equation true is 17.
4. To simplify (9^3)^6 using the Power Rule of Exponents, we multiply the exponents. So we have 9^(3*6) = 9^18.
Simplified answer: 9^18
5. To generate equivalent expressions for (6^-3)^7 x 6^20 / 6^-2, we can apply the Power Rule of Exponents to each term.
For (6^-3)^7, we use the Power Rule of Exponents, which states that we multiply the exponent outside the parentheses by each exponent inside the parentheses. So we have 6^(7*(-3)) = 6^-21.
For 6^20, we keep the same exponent.
For 6^-2, we use the Power Rule of Exponents, which states that a negative exponent in the denominator becomes positive when moved to the numerator. So we have 6^2.
The resulting expression is (6^-3)^7 x 6^20 / 6^-2 = 6^-21 x 6^20 / 6^2.
To simplify further, we subtract the exponents in the numerator and denominator. So we have 6^(-21+20-2) = 6^-3.
Simplified answer: 6^-3 (as an improper fraction) is 1/6^3 or 1/216.
1. To simplify the expression 6^10 x 6^5 using the Product Rule of Exponents, we add the exponents and keep the base the same. The rule states that a^m x a^n = a^(m + n). Therefore,
6^10 x 6^5 = 6^(10 + 5) = 6^15. Answer: 6^15.
2. To find the numerical equivalent of 8^9 x 8^-6, we use the Product Rule of Exponents. The rule states that a^m x a^n = a^(m + n), and a^m x a^-n = a^(m - n), where a ≠ 0. In this case, we have 8^9 x 8^-6, which can be simplified as 8^(9 + (-6)) = 8^3 = 512. Answer: 512.
3. To find the value of x that makes the equation 5^x ÷ 5^5 = 5^12 true, we can use the Quotient Rule of Exponents. The rule states that a^m ÷ a^n = a^(m - n), where a ≠ 0. In this case, we have 5^x ÷ 5^5 = 5^12. Applying the rule, we get 5^(x - 5) = 5^12. Since the bases are the same, the exponents must also be equal. Therefore, x - 5 = 12. Solving for x, we find x = 17. Answer: x = 17.
4. To simplify (9^3)^6 using the Power Rule of Exponents, we multiply the exponents. The rule states that (a^m)^n = a^(m * n), where a ≠ 0. In this case, we have (9^3)^6, which can be simplified as 9^(3 * 6) = 9^18. Answer: 9^18.
5. To generate equivalent expressions for (6^-3)^7 x 6^20 / 6^-2 using the Properties of Integer Exponents, we can apply the quotient rule and simplify each term separately.
Starting with (6^-3)^7, we raise the base and exponent to a power by multiplying them. So we have (6^-3)^7 = 6^(-3 * 7) = 6^-21.
Next, we simplify 6^20 / 6^-2 using the quotient rule. The rule states that a^m / a^n = a^(m - n), where a ≠ 0. In this case, we have 6^20 / 6^-2 = 6^(20 - (-2)) = 6^22.
Finally, we can multiply the two simplified expressions. (6^-3)^7 x 6^20 / 6^-2 = 6^-21 x 6^22.
To multiply the bases with the same exponent, we add the exponents and keep the base the same. So 6^-21 x 6^22 = 6^(-21 + 22) = 6^1 = 6.
Answer as an improper fraction: 6 / 1.