Question

Combined Operations with Exact Numbers. Perform each computation by calculator. $$\sqrt[4]{256} \times \sqrt{49}$$

Short answer

The fourth root of 256 is 4 and the square root of 49 is 7. The product of these numbers is 28.

Step-by-step solution

Find the Fourth Root of 256

Take the fourth root of 256 using a calculator. The fourth root of a number is the number that, when multiplied by itself three times, gives the original number. In mathematical terms, if we denote the fourth root of 256 as 'a', then it must satisfy the equation: a^4 = 256.

Find the Square Root of 49

Calculate the square root of 49 with a calculator. The square root of a number is the number that, when multiplied by itself, gives the original number. So if 'b' is the square root of 49, then b^2 = 49.

Multiply the Results

After finding the fourth root of 256, which is 4 (since 4^4 = 256), and the square root of 49, which is 7 (since 7^2 = 49), multiply these two results together to obtain the final answer.

Key concepts

Fourth Root Calculation

Understanding how to calculate the fourth root of a number is crucial in mathematics, particularly in algebra and higher-level math courses. The fourth root of a number is that number which, when raised to the power of four, equals the original number. Practically, if you're looking for the fourth root of 256, you're seeking a number which, when multiplied by itself three times (a total of four times), equals 256. This number is denoted by the radical sign with a small four, \( \sqrt[4]{256} \).

To find this value without a calculator, you might factor 256 into its prime factors, which are all twos since 256 is a power of two. This would be \( 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \) or \( 2^8 \). To find the fourth root, you would split this into four groups of two: \( (2^2) \times (2^2) \times (2^2) \times (2^2) \), which simplifies to \( 4 \times 4 \times 4 \times 4 \). Thus, the fourth root of 256 is 4. This method of factorization can be applied to find the fourth roots of perfect fourth powers.

Square Root Calculation

On the other hand, the square root calculation is a more familiar concept to many students. It involves finding a number that, when multiplied by itself, produces the given number. For example, calculating the square root of 49, denoted as \( \sqrt{49} \) involves finding a number that, when squared, equals 49.

Without a calculator, this often involves knowing your multiplication tables well or using the method of prime factorization. Since 49 is a perfect square, made up of \( 7 \times 7 \), it is straightforward to see that the square root of 49 is 7. Square roots are a fundamental concept in mathematics, used frequently in solving quadratic equations, understanding geometric shapes, and working with various formulas across different disciplines.

Multiplication of Radicals

The multiplication of radicals, although it might seem complex, follows a set of rules that simplify the process. When multiplying radicals, you can multiply the numbers inside the radicals (known as radicands) together if they have the same index (the small number in the radical sign that indicates the root).

For example, \( \sqrt[a]{x} \times \sqrt[a]{y} = \sqrt[a]{xy} \) where \( a \) is the index. However, when the indices are different, as in the multiplication of a fourth root by a square root, the rule changes. Such multiplication is performed step by step. First, you evaluate each radical independently. In our case, you find the fourth root of 256, which is 4, and the square root of 49, which is 7. Afterward, you simply multiply the two results together: \( 4 \times 7 = 28 \).

The ability to multiply radicals extends beyond the basic operations and can involve the use of algebraic properties and simplification techniques, enabling one to tackle complex expressions and equations involving radicals.