Math  /  Numbers & Operations

QuestionPart 1 of 2
Plot, compare, and order the square root, cube root, and expression. (5242),13,(12)2+36\sqrt{\left(5^{2}-4^{2}\right)}, \sqrt[3]{-1},\left(\frac{1}{2}\right)^{2}+\sqrt{36}

Studdy Solution

STEP 1

1. The expressions involve square roots, cube roots, and basic arithmetic operations.
2. The goal is to simplify each expression to its numerical value for comparison and ordering.
3. The final comparison and order will be based on the real number values of each expression.

STEP 2

1. Simplify each expression individually.
2. Calculate the numerical values of each simplified expression.
3. Compare and order the numerical values.

STEP 3

Simplify the expression (5242)\sqrt{(5^2 - 4^2)}.
First, calculate 525^2 and 424^2: 52=25 5^2 = 25 42=16 4^2 = 16

STEP 4

Subtract the squares: 5242=2516=9 5^2 - 4^2 = 25 - 16 = 9

STEP 5

Take the square root of the result: 9=3 \sqrt{9} = 3

STEP 6

Simplify the expression 13\sqrt[3]{-1}.
The cube root of 1-1 is: 13=1 \sqrt[3]{-1} = -1

STEP 7

Simplify the expression (12)2+36\left(\frac{1}{2}\right)^2 + \sqrt{36}.
First, calculate (12)2\left(\frac{1}{2}\right)^2: (12)2=14 \left(\frac{1}{2}\right)^2 = \frac{1}{4}

STEP 8

Calculate the square root of 3636: 36=6 \sqrt{36} = 6

STEP 9

Add the results together: (12)2+36=14+6=14+244=254=6.25 \left(\frac{1}{2}\right)^2 + \sqrt{36} = \frac{1}{4} + 6 = \frac{1}{4} + \frac{24}{4} = \frac{25}{4} = 6.25

STEP 10

List the simplified numerical values of each expression: - (5242)=3\sqrt{(5^2 - 4^2)} = 3 - 13=1\sqrt[3]{-1} = -1 - (12)2+36=6.25\left(\frac{1}{2}\right)^2 + \sqrt{36} = 6.25

STEP 11

Compare the numerical values to order them from smallest to largest: 1,3,6.25 -1, 3, 6.25

STEP 12

Present the expressions in order based on their numerical values: 13,(5242),(12)2+36 \sqrt[3]{-1}, \sqrt{(5^2 - 4^2)}, \left(\frac{1}{2}\right)^2 + \sqrt{36}
The final ordered list of expressions is: 13,(5242),(12)2+36 \sqrt[3]{-1}, \sqrt{(5^2 - 4^2)}, \left(\frac{1}{2}\right)^2 + \sqrt{36}

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