I am trying to figure out how to evaluate 100^-3/2. I'm going by the example (64)^2/3=16. IN the example, 64^2/3 is changed into 64^1/3^2=4^2=16. But I don't get why out of all numbers it is changed into 4^2. I know 16 goes into 64 four times, but why is the answer 16 and what becomes of the 1/3? So how would this apply to 11^-3/2? Thanks for your help!
First, you'd have to adhere to the rules of priority.
The expression 642/3 is much clearer.
The 2 in the exponent requires us to square the base, and the 3 in the denominator represents (1/3) means we need to take the cube root of the base.
So whether you do ∛((64)²) or (∛64)³ will give the same answer, i.e. ∛(4096)=16 or (4)²=16. The 4 comes from the cube root of 64, or 641/3.
For 11-3/2 it is the same procedure, by following the laws of exponents:
xa+b = xa × xb
xab = (xa)b
x-a = 1/xa
x1/a = ath root of x.
Therefore
11-3/2
=1/113/2
=1/∛(11³)
=1/∛(1331)
11-3/2
=1/113/2
=1/√(11³)
=1/√(1331)
To understand how to evaluate expressions like 100^(-3/2) and 11^(-3/2), let's start by reviewing the concept of fractional exponents.
In general, if you have a number raised to a fractional exponent, you can rewrite it as a root. For example, a^(1/2) is the square root of a, a^(1/3) is the cube root of a, and so on.
Now, let's apply this concept to the example you mentioned: (64)^(2/3) = 16. Here's how it works:
Step 1: Rewrite the exponent as a root: 2/3 can be expressed as a cube root because the denominator is 3. So, (64)^(2/3) is equivalent to the cube root of 64 raised to the power of 2: ∛(64^2).
Step 2: Simplify: The cube root of 64 is 4 because 4 × 4 × 4 = 64. So, ∛(64^2) becomes 4^2.
Step 3: Evaluate: 4^2 equals 16, as you mentioned.
Now, let's apply the same logic to 100^(-3/2):
Step 1: Rewrite the exponent as a root: -3/2 can be expressed as a square root because the denominator is 2. So, 100^(-3/2) is equivalent to the square root of 100 raised to the power of -3: √(100^-3).
Step 2: Simplify: The square root of 100 is 10 because 10 × 10 = 100. So, √(100^-3) becomes 10^-3.
Step 3: Evaluate: 10^-3 is equal to 1/10^3 or 1/1000. Therefore, 100^(-3/2) is equal to 1/1000.
Similarly, you can apply these steps to evaluate 11^(-3/2):
Step 1: Rewrite the exponent as a root: -3/2 can be expressed as a square root because the denominator is 2. So, 11^(-3/2) is equivalent to the square root of 11 raised to the power of -3: √(11^-3).
Step 2: Simplify: Since the base is not a perfect square, there is no simple simplification.
Step 3: Evaluate: √(11^-3) remains as √(1/11^3) or 1/√(11^3). At this point, you can leave the expression as is, or you can evaluate the square root of 11^3 and simplify further if needed.
So, to summarize, when evaluating expressions with fractional exponents, you can rewrite the exponent as a root and then simplify and evaluate accordingly.