Which Expression Is Equivalent To $144^{\frac{3}{2}}$?

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Understanding the Problem

When dealing with exponents, it's essential to understand the properties and rules that govern them. In this case, we're given the expression $144^{\frac{3}{2}}$ and asked to find an equivalent expression. To approach this problem, we need to recall the rules of exponents, particularly the rule for raising a power to a power.

The Rule for Raising a Power to a Power

The rule states that when we raise a power to another power, we multiply the exponents. Mathematically, this can be represented as:

(am)n=am⋅n(a^m)^n = a^{m \cdot n}

Applying the Rule to the Given Expression

Using the rule for raising a power to a power, we can rewrite the given expression as:

14432=(14412)3144^{\frac{3}{2}} = (144^{\frac{1}{2}})^3

Simplifying the Expression

Now, we need to simplify the expression inside the parentheses. To do this, we'll use the rule for finding the square root of a number, which states that:

a12=aa^{\frac{1}{2}} = \sqrt{a}

Applying this rule to the expression, we get:

(14412)3=(144)3(144^{\frac{1}{2}})^3 = (\sqrt{144})^3

Evaluating the Square Root

To evaluate the square root, we need to find the number that, when multiplied by itself, gives us 144. In this case, the square root of 144 is 12, since:

12⋅12=14412 \cdot 12 = 144

Substituting the Value of the Square Root

Now that we know the value of the square root, we can substitute it back into the expression:

(144)3=(12)3(\sqrt{144})^3 = (12)^3

Evaluating the Cubed Expression

To evaluate the cubed expression, we need to multiply 12 by itself three times:

(12)3=12⋅12⋅12=1728(12)^3 = 12 \cdot 12 \cdot 12 = 1728

Conclusion

Therefore, the expression equivalent to $144^{\frac{3}{2}}$ is:

17281728

Alternative Method: Using the Rule for Exponents with Fractional Exponents

Another way to approach this problem is to use the rule for exponents with fractional exponents, which states that:

amn=amna^{\frac{m}{n}} = \sqrt[n]{a^m}

Using this rule, we can rewrite the given expression as:

14432=14432144^{\frac{3}{2}} = \sqrt[2]{144^3}

Evaluating the Expression

To evaluate the expression, we need to find the cube of 144 and then take the square root of the result:

14432=(144⋅144⋅144)2\sqrt[2]{144^3} = \sqrt[2]{(144 \cdot 144 \cdot 144)}

Simplifying the Expression

Now, we can simplify the expression by evaluating the cube of 144:

(144⋅144⋅144)=1728(144 \cdot 144 \cdot 144) = 1728

Taking the Square Root

Finally, we take the square root of the result:

17282=12⋅12=144\sqrt[2]{1728} = 12 \cdot 12 = 144

Conclusion

However, we are looking for the value of $144^{\frac{3}{2}}$, which is equivalent to $1728$.

Alternative Method: Using the Rule for Exponents with Fractional Exponents and Simplifying

Another way to approach this problem is to use the rule for exponents with fractional exponents and simplify the expression:

14432=(14412)3=(144)3=(12)3=1728144^{\frac{3}{2}} = (144^{\frac{1}{2}})^3 = (\sqrt{144})^3 = (12)^3 = 1728

Conclusion

Therefore, the expression equivalent to $144^{\frac{3}{2}}$ is:

17281728

Final Answer

The final answer is: 1728\boxed{1728}

Q: What is the rule for raising a power to a power?

A: The rule states that when we raise a power to another power, we multiply the exponents. Mathematically, this can be represented as:

(am)n=am⋅n(a^m)^n = a^{m \cdot n}

Q: How do I simplify an expression with a fractional exponent?

A: To simplify an expression with a fractional exponent, you can use the rule for exponents with fractional exponents, which states that:

amn=amna^{\frac{m}{n}} = \sqrt[n]{a^m}

Q: What is the difference between a square root and a cube root?

A: A square root is the number that, when multiplied by itself, gives us the original number. For example, the square root of 16 is 4, since:

4⋅4=164 \cdot 4 = 16

A cube root is the number that, when multiplied by itself three times, gives us the original number. For example, the cube root of 27 is 3, since:

3⋅3⋅3=273 \cdot 3 \cdot 3 = 27

Q: How do I evaluate an expression with a cubed variable?

A: To evaluate an expression with a cubed variable, you can multiply the variable by itself three times. For example, to evaluate the expression (12)3(12)^3, you would multiply 12 by itself three times:

(12)3=12⋅12⋅12=1728(12)^3 = 12 \cdot 12 \cdot 12 = 1728

Q: What is the equivalent expression for $144^{\frac{3}{2}}$?

A: The equivalent expression for $144^{\frac{3}{2}}$ is:

17281728

Q: How do I find the square root of a number?

A: To find the square root of a number, you can use the rule for finding the square root of a number, which states that:

a12=aa^{\frac{1}{2}} = \sqrt{a}

Q: What is the difference between an exponent and a power?

A: An exponent is a number that is raised to a power. For example, in the expression 232^3, the 3 is an exponent and the 2 is the base. A power is the result of raising a number to an exponent. For example, in the expression 232^3, the 8 is the power.

Q: How do I simplify an expression with multiple exponents?

A: To simplify an expression with multiple exponents, you can use the rule for raising a power to a power, which states that:

(am)n=am⋅n(a^m)^n = a^{m \cdot n}

Q: What is the equivalent expression for $144^{\frac{1}{2}}$?

A: The equivalent expression for $144^{\frac{1}{2}}$ is:

1212

Q: How do I evaluate an expression with a fractional exponent?

A: To evaluate an expression with a fractional exponent, you can use the rule for exponents with fractional exponents, which states that:

amn=amna^{\frac{m}{n}} = \sqrt[n]{a^m}

Q: What is the difference between a positive exponent and a negative exponent?

A: A positive exponent is an exponent that is greater than 0. For example, in the expression 232^3, the 3 is a positive exponent. A negative exponent is an exponent that is less than 0. For example, in the expression 2−32^{-3}, the -3 is a negative exponent.

Q: How do I simplify an expression with a negative exponent?

A: To simplify an expression with a negative exponent, you can use the rule for negative exponents, which states that:

a−m=1ama^{-m} = \frac{1}{a^m}

Q: What is the equivalent expression for $144^{-\frac{1}{2}}$?

A: The equivalent expression for $144^{-\frac{1}{2}}$ is:

112\frac{1}{12}

Q: How do I evaluate an expression with a negative exponent?

A: To evaluate an expression with a negative exponent, you can use the rule for negative exponents, which states that:

a−m=1ama^{-m} = \frac{1}{a^m}

Q: What is the difference between a rational exponent and an irrational exponent?

A: A rational exponent is an exponent that can be expressed as a fraction. For example, in the expression 2322^{\frac{3}{2}}, the 32\frac{3}{2} is a rational exponent. An irrational exponent is an exponent that cannot be expressed as a fraction. For example, in the expression 222^{\sqrt{2}}, the 2\sqrt{2} is an irrational exponent.

Q: How do I simplify an expression with a rational exponent?

A: To simplify an expression with a rational exponent, you can use the rule for rational exponents, which states that:

amn=amna^{\frac{m}{n}} = \sqrt[n]{a^m}

Q: What is the equivalent expression for $144^{\frac{3}{2}}$?

A: The equivalent expression for $144^{\frac{3}{2}}$ is:

17281728

Q: How do I evaluate an expression with a rational exponent?

A: To evaluate an expression with a rational exponent, you can use the rule for rational exponents, which states that:

amn=amna^{\frac{m}{n}} = \sqrt[n]{a^m}

Q: What is the difference between a real number and an imaginary number?

A: A real number is a number that can be expressed as a fraction or a decimal. For example, in the expression 232^3, the 2 is a real number. An imaginary number is a number that cannot be expressed as a fraction or a decimal. For example, in the expression i3i^3, the ii is an imaginary number.

Q: How do I simplify an expression with an imaginary number?

A: To simplify an expression with an imaginary number, you can use the rule for imaginary numbers, which states that:

i2=−1i^2 = -1

Q: What is the equivalent expression for $i^3$?

A: The equivalent expression for $i^3$ is:

−i-i

Q: How do I evaluate an expression with an imaginary number?

A: To evaluate an expression with an imaginary number, you can use the rule for imaginary numbers, which states that:

i2=−1i^2 = -1

Q: What is the difference between a complex number and a real number?

A: A complex number is a number that can be expressed as a combination of real and imaginary numbers. For example, in the expression 2+3i2 + 3i, the 2 is a real number and the 3i is an imaginary number. A real number is a number that can be expressed as a fraction or a decimal.

Q: How do I simplify an expression with a complex number?

A: To simplify an expression with a complex number, you can use the rule for complex numbers, which states that:

a+bi=a2+b2a + bi = \sqrt{a^2 + b^2}

Q: What is the equivalent expression for $2 + 3i$?

A: The equivalent expression for $2 + 3i$ is:

22+32=13\sqrt{2^2 + 3^2} = \sqrt{13}

Q: How do I evaluate an expression with a complex number?

A: To evaluate an expression with a complex number, you can use the rule for complex numbers, which states that:

a+bi=a2+b2a + bi = \sqrt{a^2 + b^2}

Q: What is the difference between a polynomial and a rational expression?

A: A polynomial is an expression that consists of variables and coefficients, and the variables are raised to non-negative integer powers. For example, in the expression 2x2+3x+12x^2 + 3x + 1, the 2x^2, 3x, and 1 are polynomials. A rational expression is an expression that consists of a fraction with polynomials in the numerator and denominator.

Q: How do I simplify a polynomial?

A: To simplify a polynomial, you can use the rule for polynomials, which states that:

a+bi=a2+b2a + bi = \sqrt{a^2 + b^2}

Q: What is the equivalent expression for $2x^2 + 3x + 1$?

A: The equivalent expression for $2x^2 + 3x + 1$ is:

22+32=13\sqrt{2^2 + 3^2} = \sqrt{13}

Q: How do I evaluate a polynomial?

A: To evaluate a polynomial, you can use the rule for polynomials, which states that:

a+bi=a2+b2a + bi = \sqrt{a^2 + b^2}

Q: What is the difference between a rational expression and a polynomial?

A: A rational expression is an expression that consists of a fraction with polynomials in the numerator and denominator. For example, in the expression 2x2+3x+1x+1\frac{2x^2 + 3x + 1}{x + 1}, the 2x^2 + 3x + 1 is a polynomial and the x + 1 is a polynomial. A polynomial is an expression that consists of variables and coefficients, and the variables are raised to non-negative integer powers.

Q: How do