Problem 19 - Algebra 2

Question

Write each expression in exponential form. $$\sqrt{7 x^{3}}$$

Step-by-Step Solution

Answer

The expression \(\sqrt{7 x^{3}}\) in exponential form is \(7^{1/2} * x^{3/2}\).

Step 1: Identify the numbers and variables in the radical

The square root covers the number 7 and the variable \(x\) cubed. We can write this as \(\sqrt{7} * \sqrt{x^{3}}\) because the product of square roots equals the square root of the product.

Step 2: Convert the square roots to exponent form

Square roots can be rewritten as a power of 1/2. This gives us \(7^{1/2} * x^{3*1/2}\). The square root of \(x^{3}\) is the same as \(x^{3/2}\) because multiplying the exponents 3 and 1/2 equals 3/2.

Step 3: Write the final answer

Writing it all together, we have the final result as \(7^{1/2} * x^{3/2}\).

Key Concepts

Square Roots

Square roots are a fundamental concept in mathematics, often symbolized by the radical sign \(\sqrt{}\). They allow us to find a number that, when multiplied by itself, gives the original number under the radical. For example, the square root of 49 is 7, because 7 multiplied by itself is 49.
However, when dealing with algebraic expressions or more complex numbers, square roots can often be rewritten to make calculations easier. One way to simplify square roots is by converting them to exponential form, where the power of 1/2 represents the square root. This is particularly useful when working with variables and exponents, as it aligns with exponent rules and simplifies multiplication and division of powers.
Recognizing that \(\sqrt{a} = a^{1/2}\) is key to understanding how to manipulate expressions containing square roots.

Exponent Rules

Exponent rules are essential tools in simplifying algebraic expressions. They help guide how we calculate powers of numbers and variables. Here are some core rules that are frequently used:

• Product of Powers: \(a^m \times a^n = a^{m+n}\)
• Power of a Power: \((a^m)^n = a^{m \times n}\)
• Power of a Product: \((ab)^n = a^n \times b^n\)

Understanding these properties is crucial when converting expressions, such as square roots into exponential form, which simplifies calculations by turning radicals into manageable expressions with exponents. For instance, applying the power of a power rule simplifies \(x^{3/2}\) by showing you multiplied by the exponent \(3\) with the fraction \(1/2\).
This conversion not only eases the complexity of calculations but also highlights the relationship between various forms of representing powers.

Radicals

Radicals extend beyond square roots to include cube roots and other higher-level roots. The radical sign \(\sqrt{}\) indicates the root of a number, where a square root is specifically the second root.
When dealing with expressions such as \(\sqrt{b}\), we are looking for a number that multiplies by itself to make \(b\). Similarly, a cube root, denoted by \(\sqrt[3]{c}\), is the number when multiplied three times (itself) results in \(c\).
The process of converting radicals to exponential form is an important skill in simplifying expressions. By expressing \(\sqrt{7x^3}\) as \(7^{1/2} \times x^{3/2}\), you see not only the product of the individual components but also an adherence to consistent mathematical principles across various operations.

Variables

Variables are symbols used to represent numbers in algebraic expressions. Most commonly, these are letters like \(x\), \(y\), and \(z\). They allow us to create formulas and equations that describe a wide range of possible relationships in mathematical terms.
In expressions involving radicals and exponents, such as \(\sqrt{x^3}\), the variables play a significant role. The variable \(x\) is raised to a power (in this case, 3) and then taken its square root, transforming the expression into an exponent form \(x^{3/2}\).
The variable signifies a potentially infinite number of possibilities, giving the expression flexibility. Understanding variables within exponential or radical frameworks helps uncover the deeper algebraic relationships at play, providing insight into how these expressions work together.