Sure, let's explore the given equation step-by-step and justify it by using properties of exponents.
The given equation is:
[tex]$
(\sqrt[3]{27})(\sqrt[3]{27}) = \sqrt{81}
$[/tex]
Step 1: Simplify the cube roots and square roots using exponents
1. Cube Root Simplification:
- The cube root of a number [tex]\(x\)[/tex] can be expressed using exponents as [tex]\(x^{1/3}\)[/tex].
- Therefore, [tex]\( \sqrt[3]{27} = 27^{1/3} \)[/tex].
2. Square Root Simplification:
- Similarly, the square root of a number [tex]\(x\)[/tex] can be expressed as [tex]\(x^{1/2}\)[/tex].
- Therefore, [tex]\( \sqrt{81} = 81^{1/2} \)[/tex].
Step 2: Simplify Left Side of the Equation
We now simplify the left side of the equation:
[tex]$
(\sqrt[3]{27})(\sqrt[3]{27}) = (27^{1/3})(27^{1/3})
$[/tex]
Using the property of exponents that states [tex]\( a^m \cdot a^n = a^{m+n} \)[/tex]:
[tex]$
(27^{1/3})(27^{1/3}) = 27^{1/3 + 1/3} = 27^{2/3}
```
Step 3: Evaluate the Right Side of the Equation
We now simplify the right side of the equation:
$[/tex]
\sqrt{81} = 81^{1/2}
[tex]$
Step 4: Comparison
We need to check if both sides of the equation yield the same value. We will first state the exponents and their resultant values:
- Left side: \( 27^{2/3} = 8.999999999999998 \)
- Right side: \( 81^{1/2} = 9.0 \)
Step 5: Checking for Equality
Comparing:
`27^{2/3} = 8.999999999999998` (approximately 9)
`81^{1/2} = 9.0`
Since \( 8.999999999999998 \) is very close to 9.0, mathematically they should be equal but due to floating-point precision, there is a small difference. However, ideally:
`27^{2/3} = 81^{1/2}`
Therefore, we have justified the equation using properties of exponents.
In another approach, we can directly calculate and compare the numerical values of each term:
- The cube root of 27, \( 27^{1/3} \), equals 3.
- Squaring 3 gives us \( 3^2 = 9 \).
- Simplifying \( 81^{1/2} \) also results in 9.
Thus, numerically verifying that:
$[/tex]
(\sqrt[3]{27})(\sqrt[3]{27}) = \sqrt[3]{27^2} = \sqrt{81} = 9
[tex]$[/tex]
Conclusively, both sides of the given equation are equal based on these steps and the properties of exponents.
