Select the correct answer from each drop-down menu. The simplified form of [tex]$\frac{5 \sqrt{2} \pi-4 \sqrt{2} \pi}{\sqrt{2}}$[/tex] is $\square$ . Its estimated value is $\square$.
Answer (2)
Simplify the numerator: 5 2 π − 4 2 π = 2 π .
Rewrite the expression: 2 2 π .
Cancel the common factor: 2 2 π = π .
Estimate the value: π ≈ 3.14 . The simplified form of the expression is π , and its estimated value is approximately 3.14 .
Explanation
Understanding the Problem We are asked to simplify the expression 2 5 2 π − 4 2 π and then estimate its value.
Simplifying the Numerator First, we simplify the numerator by combining like terms: 5 2 π − 4 2 π = ( 5 − 4 ) 2 π = 1 2 π = 2 π .
Rewriting the Expression Now, we rewrite the expression as a fraction: 2 2 π .
Canceling Common Factors Next, we cancel the common factor of 2 from the numerator and the denominator: 2 2 π = π .
Estimating the Value The simplified form of the expression is π . Now we need to estimate the value of π . A common approximation for π is 3.14.
Final Answer Therefore, the simplified form of the expression is π , and its estimated value is approximately 3.14.
Examples
Understanding and simplifying expressions like this is useful in many areas of math and science. For example, when calculating the area or circumference of a circle, you often encounter π . Simplifying expressions involving π can make these calculations easier. Also, in physics, you might encounter similar expressions when dealing with oscillations or waves.
The simplified form of the expression is π , and its estimated value is approximately 3.14.
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