how do i simplify this;
8^-7/3
8^(-7/3) = (cube rt 8)^-7 = 2^-7 = 1/128
that's what i thought. thank you for the help!
To simplify the expression 8^(-7/3), you can follow these steps:
Step 1: Understand the Rule of Exponents
The rule of exponents states that for any positive number a and any real number b, a^(-b) is equal to 1/a^b. This means that when you have a negative exponent, you can rewrite it as the reciprocal of the base raised to the positive exponent.
Step 2: Apply the Rule of Exponents
For 8^(-7/3), we can rewrite it as 1/8^(7/3).
Step 3: Simplify the Base
To simplify the base 8^(7/3), we need to rewrite it in a more familiar form. Since 8 is the cube of 2 (2 * 2 * 2 = 8), we can rewrite 8 as 2^3. So, we have (2^3)^(7/3).
Step 4: Apply the Power of a Power Rule
The power of a power rule states that for any positive numbers a and b, (a^b)^c is equal to a^(b * c). Using this rule, we can simplify (2^3)^(7/3) as 2^(3 * 7/3).
Step 5: Simplify the Exponent
Multiplying 3 by 7/3, we get 21/3, which simplifies to 7. Therefore, 2^(3 * 7/3) is equal to 2^7.
Step 6: Evaluate the Expression
Finally, simplify 2^7 to get the numerical value. 2^7 equals 128.
So, the simplified form of 8^(-7/3) is 1/128.