The area of a square on grid paper is 5². Evaluate the area, draw the square, and label its area and side length. A cube has an edge length of 4 cm. Express its volume in repeated... The area of a square on grid paper is 5². Evaluate the area, draw the square, and label its area and side length. A cube has an edge length of 4 cm. Express its volume in repeated multiplication form and in exponential form, then evaluate. Arrange the following numbers in ascending order: 43, 72, 3², 9². Rewrite each power in the following products in repeated multiplication form: a) 32 × 54, b) (–3)³ × 26. Write each expression in parentheses as a power, then write the entire expression as a single power: a) (2 × 2 × 2) × (2 × 2), b) (4 × 4)(4 × 4 × 4 × 4). Write each expression in repeated multiplication form, and then as a single power: a) (–5)² × (–5)⁵, b) (3²)⁴.
Understand the Problem
The question is asking about various mathematical concepts related to exponents. It includes tasks such as evaluating areas, expressing volumes in repeated multiplication form, arranging numbers, and rewriting expressions as powers.
Answer
- Area: \( 25 \, \text{cm}^2 \) - Volume: \( 4^3 = 64 \, \text{cm}^3 \) - Ordered numbers: \( -34, 9, 25, 43, 72 \) - Powers: - a) \( 3^2 = 3 \times 3 \) - b) \( 4^6 \) - Simplified forms: - a) \( (-5)^7 \) - b) \( 3^8 \)
Answer for screen readers
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Area of square: ( 25 , \text{cm}^2 )
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Volume of cube in repeated multiplication form: ( 4 \times 4 \times 4 )
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Volume of cube in exponential form: ( 4^3 )
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Ordered numbers: ( -34, 9, 25, 43, 72 )
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Rewriting powers:
- a) ( 3^2 = 3 \times 3 )
- b) ( (-3)^3 = -3 \times -3 \times -3 )
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Single power:
- a) ( 2^5 )
- b) ( 4^6 )
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Simplifying:
- a) ( (-5)^7 )
- b) ( 3^8 )
Steps to Solve
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Calculating the area of the square First, we know the area ( A ) of a square is given by the formula: $$ A = s^2 $$ where ( s ) is the side length. Here, ( s = 5 ) cm, so: $$ A = 5^2 = 25 , \text{cm}^2 $$
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Expressing the volume of the cube in repeated multiplication form The volume ( V ) of a cube is calculated by: $$ V = s^3 $$ For a cube with edge length ( 4 , \text{cm} ): $$ V = 4^3 = 4 \times 4 \times 4 $$
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Rewriting the volume in exponential form This can also be expressed in exponential form: $$ V = 4^3 $$
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Arranging numbers in ascending order The numbers to arrange are: $$ 43 , 72 , -34 , 9 , 25 $$ Converted to numerical values:
- ( 43 ) (positive)
- ( 72 ) (positive)
- ( -34 ) (negative)
- ( 9 ) (positive)
- ( 25 ) (positive)
Arranging these gives: $$ -34, 9, 25, 43, 72 $$
- Rewriting powers in repeated multiplication For part 11: a) $$ 3^2 = 3 \times 3 $$
b) $$ (-3)^3 = -3 \times -3 \times -3 $$
- Writing expressions in parentheses as powers For item 12: a) $$ (2 \times 2 \times 2) \times (2 \times 2) = 2^5 $$
b) $$ (4 \times 4)(4 \times 4 \times 4 \times 4) = 4^6 $$
- Writing expressions in repeated multiplication form For item 13: a) $$ (-5)^2 \times (-5)^5 = (-5)^{2+5} = (-5)^7 $$
b) $$ (3^2)^4 = 3^{2 \cdot 4} = 3^8 $$
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Area of square: ( 25 , \text{cm}^2 )
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Volume of cube in repeated multiplication form: ( 4 \times 4 \times 4 )
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Volume of cube in exponential form: ( 4^3 )
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Ordered numbers: ( -34, 9, 25, 43, 72 )
-
Rewriting powers:
- a) ( 3^2 = 3 \times 3 )
- b) ( (-3)^3 = -3 \times -3 \times -3 )
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Single power:
- a) ( 2^5 )
- b) ( 4^6 )
-
Simplifying:
- a) ( (-5)^7 )
- b) ( 3^8 )
More Information
The area of the square and volume of the cube demonstrate fundamental principles of geometry. The exercises on rewriting powers and ordering numbers solidify the understanding of exponents and their properties.
Tips
- Confusing the area formula: Always remember that for a square, area is side length squared.
- Miscalculating the cube's volume: Ensure you multiply the side length three times.
- Forgetting negative signs in powers: Pay attention to whether the base is negative.
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