Question

\(\left(5^{1 / 4}\right)+6\left(405^{1 / 4}\right)\)

Short answer

The result of \(\left(5^{1 / 4}\right)+6\left(405^{1 / 4}\right)\) is approximately 34.015.

Step-by-step solution

Evaluate the Exponential Fractions

Start by calculating the value of \(5^{1 / 4}\) and \(405^{1 / 4}\). The exponent 1/4 means finding the fourth root of a number. You can use a calculator to find these values. For \(5^{1 / 4}\), it approximately equals to 1.495 and \(405^{1 / 4}\) gives the approximate value 5.42.

Multiply and Add

Next, multiply the value of \(405^{1 / 4}\) by 6. That gives \(5.42 * 6 = 32.52\). Now add the obtained value to the value of \(5^{1 / 4}\). That's \(32.52 + 1.495 = 34.015\)

Final Result

After performing the above operations, we get the final result as 34.015. Hence, \(\left(5^{1 / 4}\right)+6\left(405^{1 / 4}\right)\) is approximately equal to 34.015.

Key concepts

Exponents

Exponents are a way to represent repeated multiplication of a base number. For instance, in \(5^3\), 5 is the base and 3 is the exponent, meaning 5 is multiplied by itself 3 times: \(5 \times 5 \times 5\). Exponents make complex calculations simple by condensing multiple multiplications into a single expression. They can be positive, negative, or zero.

• Positive exponents indicate regular multiplication.
• Negative exponents suggest multiplying by the reciprocal, such as \(5^{-3} = \frac{1}{5^3}\).
• A zero exponent means the result is always 1 (e.g., \(5^0 = 1\)).

Understanding how to manipulate exponents is crucial for working with radical and fractional expressions.

Roots

Roots are the inverse operation of taking powers or exponents. When you see the symbol \(\sqrt{}\), it indicates the square root, but the concept extends to any degree of root. In the original exercise, you encounter the fourth root, denoted by \(x^{1/4}\).Finding the fourth root of a number means determining which number, when raised to the power of four, results in the original number. For instance, if we take \(81^{1/4}\), we solve for what number multiplies by itself four times to make 81. This process can sometimes require a calculator if estimating by hand is challenging.

Fractional Exponents

Fractional exponents provide another way to express roots in mathematical equations. For example, \(x^{1/2}\) signifies the square root of \(x\), and \(x^{1/3}\) represents the cube root. The fractional exponent \(x^{1/4}\) means the fourth root.Using fractional exponents is helpful because they allow for more straightforward manipulation within equations, linearizing the complexity of radicals. Converting between roots and fractional exponents involves treating the denominator of the fraction as the type of root:

• \(x^{1/n} = \sqrt[n]{x}\)
• \(\sqrt[4]{5} = 5^{1/4}\)

This relationship makes calculations simpler, especially when combining powers and roots, as seen in operations involving expressions like \(5^{1/4}\).

Mathematical Operations

Mathematical operations with radical expressions often involve using properties of exponents and roots together. In the original exercise, you first evaluate the fractional terms \(5^{1/4}\) and \(405^{1/4}\) separately.Following this, multiplication and addition are carried out. When performing mathematical operations:

• Ensure each term is simplified using correct exponent rules.
• Multiply terms as needed (e.g., \(6 \times 405^{1/4}\)).
• Add the results together to reach the final value.

This step-by-step approach allows you to handle complex radical expressions methodically, ensuring accuracy in calculations. With practice, these operations become intuitive, making solving such equations straightforward.