Sure! Let's break down the given expression step by step to understand how the given result is obtained.
We start with the expression:
[tex]\[ 15 a^{\frac{2}{3}} \cdot b^{-\frac{3}{2}} \cdot c^{\frac{1}{2}} \][/tex]
### Step 1: Understand the exponents
- [tex]\( a^{\frac{2}{3}} \)[/tex] means that [tex]\( a \)[/tex] is raised to the power of [tex]\( \frac{2}{3} \)[/tex].
- [tex]\( b^{-\frac{3}{2}} \)[/tex] means that [tex]\( b \)[/tex] is raised to the power of [tex]\( -\frac{3}{2} \)[/tex]. A negative exponent indicates a reciprocal, so [tex]\( b^{-\frac{3}{2}} = \frac{1}{b^{\frac{3}{2}}} \)[/tex].
- [tex]\( c^{\frac{1}{2}} \)[/tex] means that [tex]\( c \)[/tex] is raised to the power of [tex]\( \frac{1}{2} \)[/tex]. An exponent of [tex]\( \frac{1}{2} \)[/tex] indicates a square root, so [tex]\( c^{\frac{1}{2}} = \sqrt{c} \)[/tex].
### Step 2: Substitute the exponent values in decimal form
To match the given result closely, we can express the fractional exponents in decimal form:
- [tex]\( \frac{2}{3} \approx 0.666666666666667 \)[/tex]
- [tex]\( -\frac{3}{2} = -1.5 \)[/tex]
- [tex]\( \frac{1}{2} = 0.5 \)[/tex]
### Step 3: Rewrite the expression with decimal exponents
[tex]\[ 15 a^{0.666666666666667} \cdot b^{-1.5} \cdot c^{0.5} \][/tex]
### Step 4: Apply the negative exponent property
Rewrite [tex]\( b^{-1.5} \)[/tex] as [tex]\( \frac{1}{b^{1.5}} \)[/tex]:
[tex]\[ 15 a^{0.666666666666667} \cdot \frac{1}{b^{1.5}} \cdot c^{0.5} \][/tex]
### Step 5: Simplify the expression
Combine all the terms together:
[tex]\[ 15 \cdot a^{0.666666666666667} \cdot c^{0.5} \cdot \frac{1}{b^{1.5}} \][/tex]
This can be written in a more compact form:
[tex]\[ 15 a^{0.666666666666667} c^{0.5} / b^{1.5} \][/tex]
### Final Result:
[tex]\[ 15 a^{0.666666666666667} c^{0.5} / b^{1.5} \][/tex]
So, the detailed step-by-step solution for the expression [tex]\( 15 a^{\frac{2}{3}} \cdot b^{-\frac{3}{2}} \cdot c^{\frac{1}{2}} \)[/tex] results in:
[tex]\[ 15 \cdot a^{0.666666666666667} \cdot c^{0.5} / b^{1.5} \][/tex]
