Math Problem Statement

Solve the following questions based on square and cube numbers.

Solution

Let's go through each question based on what’s visible in the image.

Question 5

You need to compare the following expressions using <<, >>, or ==:

  1. 535^3 and 575^7

    • 53=1255^3 = 125
    • 57=78,1255^7 = 78,125
    • 53<575^3 < 5^7
  2. 16\sqrt{16} and 44

    • 16=4\sqrt{16} = 4
    • 16=4\sqrt{16} = 4, so 4=44 = 4
  3. 77297^{729} and 99

    • 77297^{729} is a very large number, much larger than 99, so 7729>97^{729} > 9
  4. 434^3 and 343^4

    • 43=644^3 = 64
    • 34=813^4 = 81
    • So 43<344^3 < 3^4
  5. 0.000170.00017 and 1.7×1051.7 \times 10^{-5}

    • 0.00017=1.7×1040.00017 = 1.7 \times 10^{-4}, which is larger than 1.7×1051.7 \times 10^{-5}, so 0.00017>1.7×1050.00017 > 1.7 \times 10^{-5}
  6. 4096\sqrt{4096} and 18318^3

    • 4096=64\sqrt{4096} = 64
    • 183=583218^3 = 5832
    • So 4096<183\sqrt{4096} < 18^3

Question 6

Sigurður has a box with a volume of 1331cm31331 \, \text{cm}^3. You are asked to find the side lengths if the box is a cube.

  • The side length of a cube with volume V=s3V = s^3 can be found as s=13313=11s = \sqrt[3]{1331} = 11.
  • So, the side lengths of the box are 11cm11 \, \text{cm}.

Question 7

The question asks why you can't take the square root of a negative number and whether it’s possible to find the cube root of a negative number.

  • Square root of a negative number: In the real number system, you cannot take the square root of a negative number because no real number squared gives a negative result.
  • Cube root of a negative number: It is possible to take the cube root of a negative number, as the result will be negative. For example, 83=2\sqrt[3]{-8} = -2.

Question 8

You’re asked to sort the numbers into the left circle for square numbers and the right circle for cube numbers. Any numbers that are both square and cube should go in the overlapping region.

Given numbers: 4,8,9,16,27,36,49,64,100,1254, 8, 9, 16, 27, 36, 49, 64, 100, 125.

  • Square numbers: 4,9,16,36,49,64,1004, 9, 16, 36, 49, 64, 100
  • Cube numbers: 8,27,64,1258, 27, 64, 125
  • Both square and cube: 6464

So:

  • Left circle: 4,9,16,36,49,1004, 9, 16, 36, 49, 100
  • Right circle: 8,27,1258, 27, 125
  • Overlap: 6464

Would you like additional details or have any questions?

Related Questions:

  1. What is the general method for comparing two numbers raised to different powers?
  2. How do we calculate higher roots, like cube roots, manually?
  3. Why does the square root of a positive number always have two solutions in real numbers?
  4. What other types of numbers (e.g., fourth powers) could fit into a Venn diagram similar to Question 8?
  5. How can we approximate values like 18318^3 without a calculator?

Tip:

For cube roots, look for perfect cubes like 8,27,64,125,8, 27, 64, 125, etc., as these simplify calculations quickly.

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Math Problem Analysis

Mathematical Concepts

Square Numbers
Cube Numbers
Inequality Symbols
Square Roots
Cube Roots

Formulas

Square root of x: √x
Cube root of x: ∛x
Comparison of powers: a^m vs b^n

Theorems

Square root and cube root properties
Properties of inequalities for powers

Suitable Grade Level

Grades 7-9