Rewrite The Following Expression In Radical Form And Evaluat

Rewrite The Following Expression In Radical Form And Evaluat

Question 1: Rewrite the following expression in radical form and evaluate.

Question 2: Simplify the expression below. Show all of your work. Do not just write the answer.

Question 3: Find the sum. Show all of your work, your answer should be in simplified radical form. 3√27 + 2√3 - √218

Question 4: Simplify the expression below. Show all of your work. 3√2x^4y^6zx^4y^6z^5

Question 5: Find the product: 2√7 × √7

Question 6: Find the quotient, show all of your work.

Question 7: Rationalize the denominator.

Question 8: Rationalize the denominator, show all of your work: 2 + 3 / (1 + √3) + √2

Question 9: Simplify: x²y⁴ / 9x⁻⁶ × (1/2)x² y⁴ / 9x - (Answer should be completely simplified and written with positive exponents only)

Question 10: Rewrite in radical form and evaluate: 27^(−1/3)

Paper For Above instruction

The process of rewriting mathematical expressions in radical form and simplifying or evaluating them is fundamental in algebra and higher mathematics. This paper explores various types of radical expressions, their transformations, and simplifications, providing step-by-step solutions to ensure clear understanding.

In algebra, radicals are expressions involving roots, typically square roots, cube roots, etc. Rewriting expressions in radical form is often necessary to simplify complex radicals or to perform operations like addition, subtraction, multiplication, and division more straightforwardly. For example, rewriting a fractional exponent as a radical helps identify simplification paths and evaluate expressions explicitly.

Furthermore, radical expressions are essential in various mathematical contexts, such as solving quadratic equations, simplifying algebraic fractions, or analyzing functions in calculus. Recognizing when and how to convert between exponential and radical forms enhances problem-solving skills and mathematical fluency.

The following solutions detail the procedures for rewriting, simplifying, and evaluating the given expressions step by step:

Question 1: Rewrite the following expression in radical form and evaluate.

Suppose the expression is a fractional exponential like 8^(2/3). To rewrite in radical form, we express it as a root and an exponent: the denominator of the fractional power indicates the root, and the numerator indicates the power. For example, 8^(2/3) = (³√8)².

Calculating this, first find the cube root of 8, which is 2, then square that result: 2² = 4. Thus, 8^(2/3) = 4.

Question 2: Simplify the expression below.

Without the actual expression, the general approach involves applying the properties of radicals and exponents—combining like radicals, simplifying radicals and coefficients, and eliminating radicals from denominators when appropriate.

Question 3: Find the sum. 3√27 + 2√3 - √218

Break down each radical:

- √27 = √(9×3) = 3√3,

- 3√27 = 3×3√3 = 9√3.

- √218 cannot be simplified further as 218 factors as 2×109, and 109 is prime, so √218 remains as is.

Thus, the sum becomes:

9√3 + 2√3 - √218 = (9√3 + 2√3) - √218 = 11√3 - √218.

Question 4: Simplify 3√2x⁴ y⁶ z x⁴ y⁶ z⁵

Combine like terms under radicals:

- x⁴ and y⁶ are perfect squares, so their roots can be taken out.

- z¹ and z⁵ combine to z⁶.

Expressed as radical:

3√(2x⁴ y⁶ z x⁴ y⁶ z⁵) = 3√(2 x⁴ x⁴ y⁶ y⁶ z z⁵)

= 3√(2 x⁸ y¹² z⁶)

Now, extract the perfect squares:

- x⁸ → (x⁴)²,

- y¹² → (y⁶)²,

- z⁶ → (z³)².

Thus,

= 3×x⁴ y⁶ z³ √2.

Question 5: Find the product: 2√7 × √7

Multiply the coefficients and the radicals:

2 × 1 = 2,

√7 × √7 = √(7×7) = 7.

Product: 2 × 7 = 14.

Question 6: Find the quotient, show all of your work

Suppose the division is (a√b) / (c√d). The general approach involves rationalizing the denominator if necessary, simplifying radicals, and dividing coefficients. For example, dividing √8 by √2:

(√8) / (√2) = √(8/2) = √4 = 2.

Question 7 & 8: Rationalize the denominator

To rationalize denominators involving radicals, multiply numerator and denominator by the conjugate of the denominator or by an appropriate radical form. For example:

- Rationalize (2 + 3/ (1 + √3)) by multiplying numerator and denominator by (1 - √3).

- Rationalize (2 + 3 / (1 + √3) + √2) similarly by multiplying numerator and denominator by the conjugate to eliminate radicals from the denominator.

Question 9: Simplify

Given expression: x² y⁴ / 9x^-6 × (1/2) x² y⁴ / 9x.

Rewrite with positive exponents:

- x^-6 = 1 / x⁶.

Simplify numerator and denominator step by step, combining like terms:

x² / x^-6 = x² × x⁶ = x⁸.

Similarly, (x²) / x = x² / x¹ = x¹ = x.

- y⁴ / y⁴ = 1.

- Include the (1/2) factor, and simplify all parts for a fully positive exponent, simplified form.

Question 10: Rewrite in radical form and evaluate: 27^{−1/3}

Using properties of exponents:

27^{−1/3} = 1 / 27^{1/3} = 1 / (³√27) = 1 / 3.

Conclusion

Mastering the conversion of exponential expressions into radical form, along with simplifying radicals and rationalizing denominators, is essential for advanced algebra and calculus. These techniques enable mathematicians and students to manipulate complex expressions effectively, facilitating problem-solving and deeper understanding of mathematical relationships.

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