Problem 34 - Intermediate Algebra

Question

Simplify each radical expression. All variables represent positive real numbers. $$ -\sqrt[7]{256 t^{6}} $$

Step-by-Step Solution

Answer

The simplified expression is \(-2 \sqrt[7]{2 t^6}\).

Step 1: Identify the Expression

The given expression is \(-\sqrt[7]{256 t^{6}}\). Our goal is to simplify this radical expression.

Step 2: Simplify the Radical

First, let's focus on simplifying the expression inside the radical, \(256 t^6\). Notice that 256 is a power of 2, specifically, \(256 = 2^8\).

Step 3: Recognize Powers and Roots

Since we have a 7th root, we aim for powers that are multiples of 7. However, \(2^8\) can be expressed as \(2^{7+1}\) or \(2^7 \times 2^1\). Thus, we can write the expression inside the radical as \(2^8 t^6 = 2^7 \times 2^1 \times t^6\).

Step 4: Simplify the Radical Expression

Now apply the 7th root to \(2^7\), which simplifies to \(2\), as \(\sqrt[7]{2^7} = 2\).Also, note that \(t^6\) remains under the radical because 6 is less than the root index 7.

Step 5: Combine the Results

Having simplified inside the radical, the expression transforms to:\(-\sqrt[7]{2} \times \sqrt[7]{t^6}\)which can be rewritten as:\(-2 \sqrt[7]{2 t^6}\).

Step 6: Final Simplified Expression

We cannot simplify further since neither \(2\) nor \(t^6\) under the 7th root are perfect powers suitable for further simplification. Thus, the simplified expression is:\(-2 \sqrt[7]{2 t^6}\).

Key Concepts

Seventh Root

The concept of taking a root, such as the seventh root, involves the idea of finding a value that, when raised to a particular power, yields the original number. In simple terms, the seventh root of a number, say "n," is a value "m" such that when "m" is multiplied by itself seven times, it equals "n." For instance, to find the seventh root of a number like 128, you are essentially asking what number multiplied by itself 7 times results in 128.

When looking at our expression, \(-\sqrt[7]{256 t^6}\), we aim to simplify inside the seventh root. We see that we can break down 256 because it is a power of 2. Therefore, recognizing these powers helps to simplify expressions rooted to the seventh degree.

Powers and Roots

Understanding powers and roots is vital in simplifying radical expressions. A power like \(256 = 2^8\) represents how many times a number is multiplied by itself. In other words, 256 is formed by multiplying the number 2, eight times together.

• Powers: A power refers to the number of times a number is multiplied by itself. For example, \(2^3 = 2 \times 2 \times 2 = 8\).
• Roots: This operation is essentially the inverse of power. The goal is to find a number, when repeatedly multiplied by itself, gives a specified number.

In our step-by-step solution, we identified that 256 can be expressed as \(2^8 = 2^7 \times 2^1\). This insight helped us manage the expression \-\sqrt[7]{256 t^6}\ to separate the factors where the seventh root could easily apply, particularly to simplify \(2^7\).

Positive Real Numbers

When dealing with radical expressions, especially when they involve variables, it's important to note that all given variables are assumed to be positive real numbers. This assumption simplifies calculations and avoids complex situations arising from negative bases and even root indices.

• Real Numbers: These are numbers that include all the rational numbers, such as fractions and integers, and all the irrational numbers.
• Positive Real Numbers: These are real numbers greater than zero. They do not include negative numbers or zero.

In the context of our expression \-\sqrt[7]{256 t^6}\, knowing that \(t\) is a positive real number ensures that operations involving \(t^6\) are straightforward. The assumption of positivity allows all squared or rooted expressions to remain defined and manageable, eliminating the need to deal with any negative root byproducts.