Problem 10 - Algebra and Trigonometry

Question

Write each radical expression using exponents, and each exponential expression using radicals. $$ 11^{-3 / 2} $$

Step-by-Step Solution

Answer

The expression \( 11^{-3/2} \) is equal to \( \frac{1}{(\sqrt{11})^3} \) in radical form.

Step 1: Understand the Negative Exponent

The negative exponent indicates a reciprocal, so we first handle the negative part of the exponent. \( 11^{-3/2} \) can be rewritten as \( \frac{1}{11^{3/2}} \). We now just focus on simplifying \( 11^{3/2} \).

Step 2: Convert to a Radical Expression

Recall that an exponent in the form of \( b^{m/n} \) represents a radical expression. The denominator \( n \) of the exponent becomes the root, and the numerator \( m \) becomes the power under the radical. So, \( 11^{3/2} \) is equivalent to \( (\sqrt{11})^3 \).

Step 3: Combine the Steps

Now, substitute the radical form back into the reciprocal form from Step 1. We have \( \frac{1}{11^{3/2}} \), which translates to \( \frac{1}{(\sqrt{11})^3} \). Therefore, the radical expression of \( 11^{-3/2} \) is \( \frac{1}{(\sqrt{11})^3} \).

Key Concepts

Exponents

Exponents are a fundamental concept in mathematics, crucial for understanding exponential growth, scientific notation, and even complex calculations. An exponent refers to the number of times a number, known as the base, is multiplied by itself. For instance, in the expression \( 2^3 \), 2 is the base and 3 is the exponent, which means \( 2 \times 2 \times 2 \). The result is 8.

In general, expressions in the form \( b^n \) are composed of:

• \( b \): the base
• \( n \): the exponent

This concept extends to different types of numbers, including integers, fractions, and radicals. Moreover, it applies not only to whole numbers but also to fractions, which we'll explore further.

Understanding exponents is vital since they simplify multiplication processes significantly and are essential in various fields such as physics, engineering, and computer science.

Negative Exponent

A negative exponent allows mathematicians to express small values using big numbers, introducing the concept of reciprocal. When you see a negative exponent like \( a^{-n} \), it means to take the reciprocal of the base raised to the positive exponent. Thus, \( a^{-n} = \frac{1}{a^n} \).

This idea is very handy for simplifying expressions and solving mathematical equations. Let's take an example with \( 11^{-3/2} \). The negative exponent here signals to take the reciprocal:

• \( 11^{-3/2} \) becomes \( \frac{1}{11^{3/2}} \).

Negative exponents may seem tricky at first, but it's merely about flipping the base and changing the sign of the exponent. This forms a crucial step in simplifying mathematical expressions efficiently.

Reciprocal

The reciprocal of a number is simply turning it into its inverse, which is easily expressed as \( \frac{1}{a} \) for any number \( a \). If you've ever tried dividing one by a number, you've encountered the reciprocal. For example, the reciprocal of 5 is \( \frac{1}{5} \).

Understanding reciprocals is key when dealing with expressions containing negative exponents. They allow us to rewrite and eventually solve complex equations.

Let's look at our example, \( 11^{-3/2} \):

• Here, utilizing reciprocals translates it into \( \frac{1}{11^{3/2}} \).

This transformation simplifies the handling of the expression as you work further into simplifying the powers or converting them into more meaningful forms, like radicals. Reciprocals turn complex multiplication into simpler divisions, fostering easier computation and clarity.

Radical Form

Radical form is another expression style for representing exponents, particularly those that involve fractions. When you express an exponent as a fraction \( b^{m/n} \), this signifies a root operation. Specifically, it means taking the \( n \)-th root of the base and raising it to the \( m \)-th power.

Consider \( 11^{3/2} \):

• The denominator (2) indicates the square root.
• The numerator (3) shows the power to which the result is raised.

Thus, \( 11^{3/2} \) can be rewritten in radical form as \( (\sqrt{11})^3 \). Transitioning between exponents and radicals makes complex expressions more flexible and manageable.

This approach is essential for simplifying expressions for calculations or understanding geometrical and physical phenomena where measurements often involve roots and powers. Radical forms find significant utility across various mathematical problems and real-world applications, solidifying our grasp of both simplicity and complexity in mathematics.