Of course! Let's solve the expression [tex]\( 2^{-3} + (0.01)^{-\frac{1}{2}} - 27^{\frac{2}{3}} \)[/tex] step-by-step.
### Step 1: Evaluate [tex]\( 2^{-3} \)[/tex]
First, we calculate [tex]\( 2^{-3} \)[/tex]. The negative exponent indicates a reciprocal, so:
[tex]\[ 2^{-3} = \frac{1}{2^3} \][/tex]
Calculating [tex]\( 2^3 \)[/tex]:
[tex]\[ 2^3 = 8 \][/tex]
So:
[tex]\[ 2^{-3} = \frac{1}{8} = 0.125 \][/tex]
### Step 2: Evaluate [tex]\( (0.01)^{-\frac{1}{2}} \)[/tex]
Next, we calculate [tex]\( (0.01)^{-\frac{1}{2}} \)[/tex]. Again, the negative exponent indicates a reciprocal, so:
[tex]\[ (0.01)^{-\frac{1}{2}} = \frac{1}{(0.01)^{\frac{1}{2}}} \][/tex]
Finding the square root of [tex]\( 0.01 \)[/tex]:
[tex]\[ \sqrt{0.01} = 0.1 \][/tex]
Taking the reciprocal of [tex]\( 0.1 \)[/tex]:
[tex]\[ \frac{1}{0.1} = 10 \][/tex]
So:
[tex]\[ (0.01)^{-\frac{1}{2}} = 10 \][/tex]
### Step 3: Evaluate [tex]\( 27^{\frac{2}{3}} \)[/tex]
Next, we calculate [tex]\( 27^{\frac{2}{3}} \)[/tex]. This expression involves both a cube root and a square:
[tex]\[ 27^{\frac{2}{3}} = \left(27^{\frac{1}{3}}\right)^2 \][/tex]
Finding the cube root of [tex]\( 27 \)[/tex]:
[tex]\[ 27^{\frac{1}{3}} = 3 \][/tex]
Squaring the result:
[tex]\[ 3^2 = 9 \][/tex]
So:
[tex]\[ 27^{\frac{2}{3}} = 9 \][/tex]
### Step 4: Combine the results
Now, we combine the results from the previous steps:
[tex]\[ 2^{-3} + (0.01)^{-\frac{1}{2}} - 27^{\frac{2}{3}} = 0.125 + 10 - 9 \][/tex]
Simplifying the expression:
[tex]\[ 0.125 + 10 - 9 = 1.125 \][/tex]
### Step 5: Final result
Thus, the value of the expression [tex]\( 2^{-3} + (0.01)^{-\frac{1}{2}} - 27^{\frac{2}{3}} \)[/tex] is [tex]\( 1.125 \)[/tex]. Also, as individual terms, we have:
- [tex]\( 2^{-3} = 0.125 \)[/tex]
- [tex]\( (0.01)^{-\frac{1}{2}} = 10 \)[/tex]
- [tex]\( 27^{\frac{2}{3}} = 9 \)[/tex]
So the detailed result is:
[tex]\[ 2^{-3} = 0.125 \][/tex]
[tex]\[ (0.01)^{-\frac{1}{2}} = 10 \][/tex]
[tex]\[ 27^{\frac{2}{3}} = 9 \][/tex]
[tex]\[ \text{Result} = 1.125 \][/tex]
I hope this detailed solution helps you understand the steps involved in evaluating the expression!
