Ask a New Question
Menu

Consider the expression 36 −−−√

x 5/3
. Will the product be rational or irrational? Explain how you know in one or two sentences, and then find the product. (2 points) put it in a format that is easily readable

11 answers

To evaluate the expression \( \sqrt{36} \times x^{5/3} \), we first simplify \( \sqrt{36} \), which is equal to \( 6 \). Since \( x^{5/3} \) can be rational or irrational depending on the value of \( x \), the product \( 6 \times x^{5/3} \) will be rational if \( x^{5/3} \) is rational, and irrational if \( x^{5/3} \) is irrational.

Expression:

\[ \sqrt{36} \cdot x^{5/3} = 6 \cdot x^{5/3} \]

Product:

\[ 6x^{5/3} \]

Thus, the rationality of the product depends on \( x \). For example, if \( x = 1 \) (rational), then the product is \( 6 \); if \( x = 2 \) (rational), then the product is \( 6 \cdot 2^{5/3} \) (which can be expressed as irrational). Therefore, the product can be either rational or irrational depending on the value of \( x \).

make this understandable

Let's break this down step by step.

Expression:

We are looking at the expression: \[ \sqrt{36} \cdot x^{5/3} \]

Step 1: Simplify \(\sqrt{36}\)

The square root of \(36\) is \(6\): \[ \sqrt{36} = 6 \]

Step 2: Rewrite the Expression

Now we can rewrite the expression as: \[ 6 \cdot x^{5/3} \]

Rational or Irrational?

  • The expression \(6 \cdot x^{5/3}\) will be rational if \(x^{5/3}\) is a rational number.
  • It will be irrational if \(x^{5/3}\) is an irrational number.

Example:

  1. If \(x = 1\):

    • \(x^{5/3} = 1^{5/3} = 1\) (rational)
    • Product: \(6 \cdot 1 = 6\) (rational)

  2. If \(x = 2\):

    • \(x^{5/3} = 2^{5/3}\) (this is not a simple rational number)
    • Product: \(6 \cdot 2^{5/3}\) (irrational)

Conclusion:

The rationality of the overall product \(6 \cdot x^{5/3}\) depends on the value of \(x\):

  • If \(x\) makes \(x^{5/3}\) rational, then the product will be rational.
  • If \(x\) makes \(x^{5/3}\) irrational, then the product will be irrational.

The final form of the expression is: \[ 6x^{5/3} \]

This simple breakdown shows how the product can be rational or irrational based on the choice of \(x\).

no, i mean make it readable. i dont know what you mean with all the weird wording you use

Got it! Let's simplify the explanation and make it clearer.

Given Expression:

We have the expression: \[ \sqrt{36} \cdot x^{5/3} \]

Step 1: Simplify \(\sqrt{36}\)

  • The square root of 36 is 6: \[ \sqrt{36} = 6 \]

Step 2: Write the Expression Again

Now, we can write the expression as: \[ 6 \cdot x^{5/3} \]

Will the Product be Rational or Irrational?

  • The product \(6 \cdot x^{5/3}\) can be either rational or irrational, depending on \(x\).

Examples:

  1. If \(x = 1\):

    • Then \(x^{5/3} = 1^{5/3} = 1\) (which is rational).
    • The product is \(6 \cdot 1 = 6\) (rational).

  2. If \(x = 2\):

    • Then \(x^{5/3} = 2^{5/3}\) (this number is not a simple rational number).
    • The product is \(6 \cdot 2^{5/3}\) (this value is irrational).

Conclusion:

  • The product \(6 \cdot x^{5/3}\) is:
    • Rational if \(x^{5/3}\) is rational.
    • Irrational if \(x^{5/3}\) is irrational.

Final Expression:

The final expression you need is: \[ 6x^{5/3} \]

That's it! The product's nature (rational or irrational) depends on the value of \(x\).

uggh, i give up. this is so stupid. you dont know what i mean

I apologize for any confusion earlier. Let’s simplify this as much as possible and focus on clarity:

Problem:

We need to determine if the expression \( \sqrt{36} \cdot x^{5/3} \) is rational or irrational.

Step 1: Calculate \( \sqrt{36} \)

  • \( \sqrt{36} = 6 \)

Step 2: Rewrite the Expression

  • The expression simplifies to: \[ 6 \cdot x^{5/3} \]

Step 3: Determine Rationality

  • Rational: If \( x^{5/3} \) is a rational number.
  • Irrational: If \( x^{5/3} \) is an irrational number.

Examples:

  1. For \( x = 1 \):

    • \( 1^{5/3} = 1 \) (rational)
    • Product: \( 6 \times 1 = 6 \) (rational)

  2. For \( x = 2 \):

    • \( 2^{5/3} \) (this can be irrational)
    • Product: \( 6 \times 2^{5/3} \) (this is often irrational)

Final Thought:

  • The product \( 6 \cdot x^{5/3} \) can be either rational or irrational depending on \( x \).

Final Expression:

The expression is: \[ 6 \cdot x^{5/3} \]

Thank you for your patience, and I hope this clearer format helps! If you have more specific requests or questions, please let me know.

can you do EXACTLY what you just did, but put everything into words?

Sure! Here’s the explanation in plain words without mathematical symbols:

We need to evaluate the expression that consists of the square root of 36 multiplied by \( x \) raised to the power of \( \frac{5}{3} \). The goal is to find out if the result is a rational number or an irrational number.

  1. First, we calculate the square root of 36.
    The square root of 36 is 6.

  2. Next, we can rewrite the expression.
    After simplifying, we see that it becomes 6 multiplied by \( x \) raised to the power of \( \frac{5}{3} \).

  3. Now, we need to understand when this product will be rational or irrational.

    • The outcome will be rational if \( x \) raised to the power of \( \frac{5}{3} \) is a rational number.
    • The outcome will be irrational if \( x \) raised to the power of \( \frac{5}{3} \) is an irrational number.

  4. Let’s look at a couple of examples to clarify this further:

    • Example 1: If we let \( x \) be 1. When we calculate \( 1 \) raised to the power of \( \frac{5}{3} \), the result is 1, which is a rational number. Therefore, when we multiply 6 by 1, the result is 6, which is also rational.

    • Example 2: If we let \( x \) be 2. When we calculate \( 2 \) raised to the power of \( \frac{5}{3} \), this result is not easily identifiable as a simple rational number. So, when we multiply 6 by \( 2 \) raised to the power of \( \frac{5}{3} \), the overall result tends to be an irrational number.

  5. In conclusion, the expression will yield varying results.
    The product can be rational or irrational depending on the value chosen for \( x \).

  6. To summarize, the final expression we are working with is 6 multiplied by \( x \) raised to the power of \( \frac{5}{3} \).

I hope this explanation is clearer for you! If you need anything else or have more questions, feel free to ask.

Consider the expression 3/4+10−−√
. Is the sum rational or irrational? Explain your answer in 1-2 sentences. (2 points)

To determine if the sum \( \frac{3}{4} + \sqrt{10} \) is rational or irrational, we need to note that \( \sqrt{10} \) is an irrational number. The sum of a rational number (\( \frac{3}{4} \)) and an irrational number (\( \sqrt{10} \)) is always irrational. Therefore, the sum \( \frac{3}{4} + \sqrt{10} \) is irrational.