Question

By giving specific values for \(a, b,\) and \(c,\) explain how the exponent rule $$ \left(a^{b}\right)^{c}=a^{b c} $$ is used to rewrite the expressions $$ \sqrt{(x+1)\left(x^{2}+2 x+1\right)}=(x+1)^{3 / 2} $$

Short answer

Question: Apply the exponent rule \((a^b)^c = a^{bc}\) and rewrite the expression \(\sqrt{(x+1)(x^2+2x+1)}\) in terms of powers of \((x+1)\). Answer: Applying the exponent rule to the expression, we obtain \((x+1)^1\), which is equivalent to \((x+1)\).

Step-by-step solution

Identify values for a, b, and c

We need to find the specific values for \(a, b,\) and \(c\) such that the given expression will follow the exponent rule. In this case, we can observe that inside the square root we have the product of \((x+1)\) and \((x^2+2x+1)\), and we are supposed to rewrite this expression as \((x+1)^{3/2}\). This means that we should be able to rewrite the expression inside the square root as \((x+1)^2\). From this observation, we can identify the values of \(a, b,\) and \(c\) as follows: - \(a = (x+1)\) - \(b = 2\) - \(c = 1/2\)

Apply the exponent rule to rewrite the expression

Using the values of \(a, b,\) and \(c\) we have identified, we can now apply the exponent rule \((a^b)^c = a^{bc}\): $$\left((x+1)^2\right)^\frac{1}{2} = (x+1)^{2 \times \frac{1}{2}}$$ Since. \(2 \times \frac{1}{2} = 1\), we have: $$\left((x+1)^2\right)^\frac{1}{2} = (x+1)^1$$

Verify the equivalence of the expressions

Now let's check the equivalence between the given expression and the simplified expression: 1. \(\sqrt{(x+1)(x^2+2x+1)}\): This expression represents the square root of the product of the two factors \((x+1)\) and \((x^2+2x+1)\). 2. \((x+1)^1\): This expression represents the factor \((x+1)\) raised to the power of 1, which is simply \((x+1)\). To verify the equivalence of these expressions, first observe that the factor \((x^2 + 2x + 1)\) is equivalent to \((x+1)^2\). Therefore, we can rewrite the first expression as: $$\sqrt{(x+1)(x+1)^2}$$ Applying the exponent rule we found earlier, this expression becomes: $$\left((x+1)^2\right)^\frac{1}{2}$$ And as we saw before, this expression simplifies to \((x+1)^1\) or simply \((x+1)\), which is equal to the simplified expression. Therefore, the given expression is equivalent to the simplified expression, verifying the use of the exponent rule.

Key concepts

Algebraic Expressions

Algebraic expressions are mathematical phrases that involve numbers, variables, and operations. In the exercise above, our expression includes several components that work together to exhibit algebraic rules, especially the exponent rule. When dealing with algebraic expressions, it's important to recognize each component and its role in the expression.

• **Variables**: In our example, the letter \(x\) is a variable, representing a number that can change.
• **Constants**: Numbers such as 1 and 2 are constants; their value doesn't change.
• **Operations**: Operations in this context include addition, multiplication, and exponentiation.

In our exercise, this knowledge helps in simplifying the expression. By recognizing that \((x^2 + 2x + 1)\) is actually a perfect square, expressed equivalently as \((x+1)^2\), we can transform and simplify the entire expression under the square root. This understanding facilitates the application of further mathematical rules, like the exponent rule.

Square Root Simplification

The square root simplification is a key process in this exercise, allowing the reduction of a more complex expression into a simpler form. The square root, \(\sqrt{}\), is essentially the reverse of squaring a number. For instance, the square root of 4 is 2, because \(2^2 = 4\).
In the context of the original exercise, we start with \(\sqrt{(x+1)(x^2+2x+1)}\). Recognizing that the polynomial \((x^2 + 2x + 1)\) is \((x+1)^2\) is crucial. Therefore, our original expression can be rewritten as \(\sqrt{(x+1) \cdot (x+1)^2}\).

• This simplification allows us to reassess the expression under the square root as a single unit: \((x+1)^3\).
• Applying the square root leads to \((x+1)^{3/2}\), as the square root of a power of 3 becomes a power of 1.5 or \(\frac{3}{2}\).

Understanding these steps reveals how algebraic manipulation of expressions involving square roots can simplify complex calculations.

Exponentiation

Exponentiation is the mathematical operation involving raising a number, the base, to the power of an exponent. When we see an expression like \((x+1)^{3/2}\), it involves exponentiation. Here,
- The base is \(x+1\), - And the exponent is \(\frac{3}{2}\).
Exponentiation follows rules that are very useful. These rules help to simplify and manipulate expressions, as demonstrated in the exercise.

• The core rule used here is \((a^b)^c = a^{bc}\). This tells us how to simplify expressions where one exponent is raised to another power.
• Another key aspect is knowing that \((a^b)^c = a^{bc}\) means we can multiply the exponents when working with powers of powers.

In practice, applying this rule streams down the process of managing complex algebraic expressions. Use of these powerful rules not only transforms how we manage expressions but it ensures results are both accurate and simplified wherever possible.