5. 8^3 * 8^7 (simplify)
11. ((5/8))^3 (simplify)
12. ((-2/9)^3)^5
16. 3^2 * 3^x = 3^12
how do I do these problems?
when multiplying powers, add the exponents.
when raising a power to a power, multiply exponents. This is because, for example,
(x^2)^3 = (x^2)(x^2)(x^2) = x^(2+2+2)
So, you have
#5: 8^3 * 8^7 = 8^(3+7) = 8^10
#11: (5/8)^3 = 5^3/8^3 = 125/512
#12: ((-2/9)^3)^5 = (-2/9)^15 = -2^15/9^15
#16
3^2 * 3^x = 3^12
3^(2+x) = 3^12
2+x = 12
x = 10
To simplify exponential expressions, we can use the properties of exponents. Let's simplify each of the given expressions step by step:
5. To simplify 8^3 * 8^7, we can add the exponents since the base (8) is the same. This rule is called the product rule of exponents. So, 8^3 * 8^7 equals 8^(3+7), which is equal to 8^10.
11. To simplify ((5/8))^3, we can cube both the numerator and the denominator. So ((5/8))^3 becomes (5^3)/(8^3) which equals 125/512.
12. To simplify ((-2/9)^3)^5, we can focus on the exponent first. The exponent outside the parentheses (5 in this case) distributes to the exponent inside the parentheses (3 in this case). So, ((-2/9)^3)^5 becomes (-2/9)^(3*5), which simplifies to (-2/9)^15.
16. To solve the equation 3^2 * 3^x = 3^12, we can use another property of exponents. When the bases are the same, we can equate the exponents. So, we have 2 + x = 12. Solving this equation for x, we subtract 2 from both sides, resulting in x = 10.