Equations with Rational Exponents Worksheet

If you're a high school student struggling to grasp equations with rational exponents, this worksheet is designed to help you solidify your understanding and improve your problem-solving skills. Focused on the concepts of rational exponents and exponent properties, this worksheet provides a variety of practice problems to challenge your knowledge and ensure you feel confident when facing similar equations in the future.

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Simplify the expression: (3^2)^(1/2)

The expression simplifies to 3.

Solve the equation: 4^(2x) = 16

To solve the equation 4^(2x) = 16, we first need to write 16 as a power of 4. Since 16 = 4^2, we can rewrite the equation as 4^(2x) = 4^2. This implies that 2x = 2. Dividing both sides by 2, we find that x = 1. Thus, the solution to the equation is x = 1.

Simplify the expression: ((5/2)^(-1/3))^2

To simplify the expression ((5/2)^(-1/3))^2, we use the rule of exponents that states when raising a power to another power, we multiply the exponents. First, we simplify the exponent of (5/2)^(-1/3) by reciprocating the fraction and changing the sign of the exponent, resulting in (2/5)^(1/3). Then, squaring this expression, we multiply the exponents 1/3 * 2, giving us the final simplified expression of (2/5)^(2/3).

Solve the equation: (2/3)^(2x - 1) = 3

To solve the equation (2/3)^(2x - 1) = 3, we first rewrite 3 as 3/1 to match the base as a fraction. Then we convert 3 to a fraction with the same denominator as 2/3, which gives 9/3. So, the equation becomes (2/3)^(2x - 1) = 9/3. Simplifying further, we get (2/3)^(2x - 1) = 3^2. Comparing the exponents on both sides, we have 2x - 1 = 2. Solving for x gives x = 3/2.

Simplify the expression: (x^(-3/4)) / (x^(1/2))

To simplify the expression (x^(-3/4)) / (x^(1/2)), we can apply the rule of subtracting exponents when dividing like bases. Simplifying further, we get x^((-3/4) - (1/2)), which is x^(-3/4 - 2/4) = x^(-5/4). Therefore, the simplified expression is x^(-5/4).

Solve the equation: 9^(2x) = 1/81

To solve the equation 9^(2x) = 1/81, we can rewrite 1/81 as 9^(-2) because 1/81 is equivalent to 9 raised to the power of -2. Now we have 9^(2x) = 9^(-2), which means 2x = -2. Simplifying further, we find x = -1. Therefore, the solution to the equation is x = -1.

Simplify the expression: (8/27)^(3/2)

To simplify the expression (8/27)^(3/2), we can first rewrite 8 as 2^3 and 27 as 3^3. This gives us (2^3/3^3)^(3/2), which can then be simplified to (2/3)^3. So, the simplified expression is (2/3)^3.

Solve the equation: 2^(x/2) = 8^(1 - x)

To solve the equation \(2^{\frac{x}{2}} = 8^{1 - x}\), note that 8 can be written as \(2^3\). So the equation becomes \(2^{\frac{x}{2}} = (2^3)^{1 - x}\), which simplifies to \(2^{\frac{x}{2}} = 2^{3 - 3x}\). Now, since the bases are the same, we can equate the exponents to get \(\frac{x}{2} = 3 - 3x\). Solving this equation yields x = 0.

Simplify the expression: ((4/9)^3)^(-1/2)

To simplify the expression ((4/9)^3)^(-1/2), first calculate (4/9)^3 which equals 64/729. Then, raise that result to the power of -1/2, which is the same as taking the square root of the reciprocal, resulting in 27/8 as the simplified expression.

Solve the equation: (1/2)^(3x - 1) = 2^(1 - 2x)

To solve the equation (1/2)^(3x - 1) = 2^(1 - 2x), we can rewrite it as (2^(-1))^(3x - 1) = 2^(1 - 2x), which simplifies to 2^(-3x + 1) = 2^(1 - 2x). Now, we can equate the exponents: -3x + 1 = 1 - 2x. Solving this equation gives x = 0. Thus, the solution to the equation is x = 0.