For question 1, the area of a square is given by side². Taking the square root of 60025, we find that √60025 = 245. Thus, the length of the field is 245 m.

For question 3, calculate each cube root individually: [3]27 = 3, [3]8 = 2, and [3]64 = 4. Adding these values together gives 3 + 2 + 4 = 9.

For question 4, factor composite bases into prime factors: 10 = 2 × 5, 125 = 5³, and 6 = 2 × 3. The common factors 3^(-5) and 2^(-5) in numerator and denominator cancel out, leaving 5^(-5 + 3 - (-7)) = 5^5 = 3125.

For question 4, factor composite bases into prime factors: 10 = 2 × 5, 125 = 5³, and 6 = 2 × 3. The common factors 3^(-5) and 2^(-5) in numerator and denominator cancel out, leaving 5^(-5 + 3 - (-7)) = 5^5 = 3125.

For question 5, since the bases on both sides of (25)²x+1 = (25)^5 are identical, equate the exponents: 2x + 1 = 5. Subtracting 1 gives 2x = 4, which simplifies to x = 2.

For question 6, expand using the distributive property: p(p² - pq + q²) + q(p² - pq + q²) = p³ - p²q + pq² + p²q - pq² + q³. The middle terms cancel out, leaving p³ + q³.

For question 7, the sum of all interior angles in any quadrilateral is 360^°. Setting up the equation x + 50^° + 130^° + 120^° = 360^° gives x + 300^° = 360^°, hence x = 60^°.

Worksheet 5 1) The area of a square field is $60025\text{ m}^2$ . Find the length of the field. 2) Express $121$ as a sum of $11$ consecutive odd numbers. 3) Evaluate: $\sqrt[3]{27} + \sqrt[3]{8} + \sqrt[3]{64}$ 4) Evaluate: $\frac{3^{-5} \times 10^{-5} \times 125}{5^{-7} \times 6^{-5}}$ 5) Find the value of $x$ : $(25)^{2x+1} = (25)^5$ 6) Multiply: $(p+q)(p^2 - pq + q^2)$ 7) Find the angle measure ' $x$ ' in a quadrilateral with interior angles $50^\circ$ , $130^\circ$ , & $120^\circ$ .

Answer: 1) $245\text{ m}$ , 2) $1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21$ , 3) $9$ , 4) $3125$ , 5) $x = 2$ , 6) $p^3 + q^3$ , 7) $x = 60^\circ$

Step-by-step solution

Step 1: Find the side length of the square field

For question 1, the area of a square is given by $\text{side}^2$ . Taking the square root of $60025$ , we find that $\sqrt{60025} = 245$ . Thus, the length of the field is $245\text{ m}$ .

Step 2: Express 121 as a sum of 11 consecutive odd numbers

For question 2, recall that the square of any natural number $n$ is equal to the sum of the first $n$ consecutive odd numbers. Since $121 = 11^2$ , it is the sum of the first $11$ odd numbers starting from $1$ .

Step 3: Evaluate the sum of cube roots

For question 3, calculate each cube root individually: $\sqrt[3]{27} = 3$ , $\sqrt[3]{8} = 2$ , and $\sqrt[3]{64} = 4$ . Adding these values together gives $3 + 2 + 4 = 9$ .

Step 4: Simplify using laws of exponents

For question 4, factor composite bases into prime factors: $10 = 2 \times 5$ , $125 = 5^3$ , and $6 = 2 \times 3$ . The common factors $3^{-5}$ and $2^{-5}$ in numerator and denominator cancel out, leaving $5^{-5 + 3 - (-7)} = 5^5 = 3125$ .

Step 5: Solve for x in the exponential equation

For question 5, since the bases on both sides of $(25)^{2x+1} = (25)^5$ are identical, equate the exponents: $2x + 1 = 5$ . Subtracting $1$ gives $2x = 4$ , which simplifies to $x = 2$ .

Step 6: Multiply the algebraic polynomials

For question 6, expand using the distributive property: $p(p^2 - pq + q^2) + q(p^2 - pq + q^2) = p^3 - p^2q + pq^2 + p^2q - pq^2 + q^3$ . The middle terms cancel out, leaving $p^3 + q^3$ .

Step 7: Find the missing angle of the quadrilateral

For question 7, the sum of all interior angles in any quadrilateral is $360^\circ$ . Setting up the equation $x + 50^\circ + 130^\circ + 120^\circ = 360^\circ$ gives $x + 300^\circ = 360^\circ$ , hence $x = 60^\circ$ .

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