Algebra Class 10: Simplifying Radicals

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Questions and Answers

What is the simplified form of $\sqrt{36a^6b^2}$?

$12a^3b$

$6a^2b$

$36ab$

$6a^3b$ (correct)

What is the result of simplifying $\frac{\sqrt{72x^4y}}{x^2}$?

$6x\sqrt{2y}$ (correct)

$3x^2\sqrt{2y}$

$6\sqrt{2}$

$12\sqrt{2y}$

Calculate the distance between the points (-5, 6) and (-3, -4).

$8$

$10$ (correct)

$12$

$14$

What is the final result of $14\sqrt{3} + 5\sqrt{2} - 5\sqrt{3} - 6\sqrt{2}$?

$11\sqrt{3} - \sqrt{2}$ Signup and view all the answers

What is the result of $(5 - \sqrt{3})(5 + \sqrt{3})$?

$22$ Signup and view all the answers

What is the simplified form of $\sqrt{121}$?

11 Signup and view all the answers

What is the result of simplifying $\sqrt{(-7)^{2}}$?

7 Signup and view all the answers

What is the simplified form of $\sqrt{(xy)^{2}}$?

$xy$ Signup and view all the answers

What is the simplified form of $\sqrt{(6x-7)^{2}}$?

$|6x - 7|$ Signup and view all the answers

What is the simplified form of $-3\sqrt{108}$?

$-18\sqrt{3}$ Signup and view all the answers

What is the result of the expression $4i(2-3i)$?

$-12 + 8i$ Signup and view all the answers

What is the product of the complex numbers $(2+3i)(5-2i)$?

$31 + 4i$ Signup and view all the answers

Which of the following is a correct statement about the function $x^{2} + 25 = 0$?

Real numbers cannot be solutions. Signup and view all the answers

When multiplying two complex numbers, $(2+3i)$ and $(5-2i)$, which operation is used?

Distributive property Signup and view all the answers

What does $i$ represent in complex numbers?

$ ext{the square root of } -1$ Signup and view all the answers

What is the simplified form of $16^{1/2}$?

4 Signup and view all the answers

What is the result of $6^{1/2} imes 6^{3/2}$?

$6^{2}$ Signup and view all the answers

How do you write $72^{1/2}$ in its simplest form?

6 Signup and view all the answers

What is the complex number $5 - rac{ ext{sqrt}(-16)}{1}$ in standard form?

5 + 4i Signup and view all the answers

What is the solution to the equation $ ext{sqrt}(x-5) + 4 = 8$?

9 Signup and view all the answers

What is the simplified form of $rac{2 ext{√}3x}{ ext{√}6xy}$?

$rac{ ext{√}x}{ ext{√}2}$ Signup and view all the answers

What is the radical form of $11^{1/2}$?

$ ext{√}11$ Signup and view all the answers

What is the expression $rac{ ext{√}3}{ ext{√}2 + 1}$ rationalized?

$rac{ ext{√}3( ext{√}2 - 1)}{1}$ Signup and view all the answers

What is the evaluation of $2^{2/3}$ in radical form?

$ ext{√[3]{2^2}}$ Signup and view all the answers

What is the simplified form of $a^{-4/5}$?

$rac{1}{ ext{√[5]{a^4}}}$ Signup and view all the answers

Study Notes

Simplifying Radicals

Understanding square roots:
- The square root of a number is a value that, when multiplied by itself, equals the original number.
- For example, the square root of 121 is 11 because 11 * 11 = 121.
Simplifying square roots:
- Find the largest perfect square that divides the radicand (the number under the radical sign).
- For example, √125 can be simplified as √(25 * 5) = √25 * √5 = 5√5.
Simplifying radicals with variables:
- Apply the rule √(a^2) = |a| for any real number a.
- For example, √(x^2) = |x|.
Simplifying radicals with fractions:
- Apply the rule √(a/b) = √a / √b (for both a and b nonnegative).
- For example, √(81/16) = √81 / √16 = 9/4.

Adding and Subtracting Radicals

Combining like radicals:
- Radicals can only be added or subtracted if they share the same radicand and index (the small number outside the radical sign).
- For example, 3√2 + 5√2 = 8√2.

Multiplying Radicals

Multiplying roots:
- Apply the rule √a * √b = √(a * b).
- For example, √6 * √15 = √(6 * 15) = √90 = √(9 * 10) = 3√10.
Squaring a radical:
- Apply the rule (√a)^2 = a.
- For example, (√7)^2 = 7.

Dividing Radicals

Dividing roots:
- Apply the rule √a / √b = √(a/b) (for both a and b nonnegative).
- For example, √72 / √2 = √(72/2) = √36 = 6.
Rationalizing the denominator:
- To eliminate radicals from the denominator, multiply both the numerator and denominator by the radical's conjugate. The conjugate of √a + √b is √a - √b.
- For example, to rationalize the denominator of √3 / (√2 + 1), multiply by (√2 - 1) / (√2 - 1).

Fractional Exponents

Understanding fractional exponents:
- A fractional exponent like x^(m/n) represents a radical expression where the numerator (m) is the power and the denominator (n) is the root.
- For example, x^(2/3) = (∛x)^2.

Complex Numbers

Understanding complex numbers:
- Complex numbers involve the imaginary unit 'i', where i² = -1.
- A complex number is in the form a + bi, where a and b are real numbers.
Operations on complex numbers:
- Addition/Subtraction: Combine real and imaginary parts separately.
- Multiplication: Use the distributive property (like multiplying binomials), remembering that i² = -1.
- Division: Rationalize the denominator by multiplying both the numerator and denominator with the complex conjugate of the denominator.
- The complex conjugate of a + bi is a - bi.

Solving Radical Equations

Isolate the radical term: Combine terms to bring the radical term to one side of the equation.
Square both sides: Eliminate the radical by squaring both sides of the equation. Remember to check for extraneous solutions, as squaring both sides can introduce them.
Solve the resulting equation: Solve the quadratic equation to find the possible values of the variable.

Distance Formula

Distance between two points: The distance between two points (x₁, y₁) and (x₂, y₂) is found using the distance formula: √((x₂ - x₁)² + (y₂ - y₁)²).

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Description

Test your knowledge on simplifying radicals, including square roots, radicals with variables, and fractions. Additionally, explore how to add and subtract like radicals. This quiz will help reinforce your understanding of these important algebraic concepts.