Question

Simplify the expression.\(\sqrt{8 x^{4} y^{3} z^{-2}}\)

Short answer

The simplified expression is \(2x^2z^{-1}\sqrt{y}\).

Step-by-step solution

Simplification Explanation

To simplify this expression, one should recall that the square root of x^2 is |x|. Therefore, when there is an exponential variable inside the square root, its exponent can be halved. However, keep in mind that the variables' exponents must be integers. So the cube term y^3 will be handled separately.

Simplify x-term and z-term

The square root of \(8x^4z^{-2}\) is \(2x^2z^{-1}\). This simplification comes from taking the square root of 8 (which is 2) and halving the exponents for x and z.

Simplify y-term

For the y term, take the square root of y^3 manually. The square root of \(y^3\) will be \(y*\sqrt{y}\). So one y comes out of the square root, and the remaining one stays inside it.

Put It All Together

Finally, combine all parts to obtain the simplified radical: \(2x^2z^{-1}*\sqrt{y}\).

Key concepts

Square Root Simplification

Understanding how to simplify square roots is a fundamental skill in algebra. The process involves finding an equivalent expression where the radical is as reduced as possible. For instance, to simplify \(\sqrt{8 x^{4} y^{3} z^{-2}}\), the first step is to break it down into prime factors and pair the variables. Take the square root of each pair, which is equivalent to dividing the exponent by two.

For perfect squares inside the radical, like \(x^4\), simplifying is straightforward because \(x^2\) is the square root of \(x^4\). However, when dealing with coefficients like 8 (which is \(2^3\)), take out the largest square factor, resulting in 2 outside the radical, leaving the remaining factor, if any, inside. In this case, 2 comes out, and there's no remainder. Numbers or variables with odd powers, like \(y^3\), are partially simplified by extracting as much as possible, leaving a single occurrence of the base underneath the radical sign.

Exponent Rules

To handle exponents when simplifying radical expressions, remember a few key exponent rules. Firstly, when you square root an exponent, you essentially halve the exponent; this follows from the rule \( (a^{m})^{n} = a^{m*n} \), where taking the square root is the same as raising to the exponent of \(1/2\).

Next, when you encounter a negative exponent, such as \(z^{-2}\), it's helpful to consider this as \(1/z^{2}\) before taking the square root, which results in \(z^{-1}\) or \(1/z\) after simplification. This is grounded in another rule calculating with negative exponents, translating them to their positive counterparts' reciprocals.

By applying these exponent rules to variables within a square root, you can significantly simplify radical expressions, transforming a complex radical into a more manageable form.

Radical Expressions in Algebra

When working with radical expressions in algebra, it's important to recognize that they can contain numbers, variables, and even other expressions within a radical sign. For the given expression \(\sqrt{8 x^{4} y^{3} z^{-2}}\), it's a mix of an integer, positive exponents, and a negative exponent all under a square root.

Radical expressions can be simplified by applying rules of arithmetic and algebra to both the radicand (the expression inside the radical) and the index (the degree of the root). It's crucial to express the radicand in its simplest form, which often means factoring numbers and using properties of exponents. This yields a product of simpler radical expressions that can be more easily managed.

Finally, remember that if a variable has an odd exponent, the radical expression will likely still contain a radical part after simplification. In our specific example, the \(y^3\) term partially simplifies, with \(y\) coming out of the square root and \(\sqrt{y}\) remaining inside. This reflects a well-rounded understanding of the processes involved in the engagement with radical expressions across different algebraic scenarios.