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Prime factorization is a method that involves breaking down a number into its basic building blocks: prime numbers. A prime number is a number greater than 1 that has no other factors except for 1 and itself. For example, the number 32 can be factored as follows:

• 32 is even, so we can divide it by 2: 32 ÷ 2 = 16. Record the factor 2.
• 16 is also even, divide by 2 again: 16 ÷ 2 = 8. Record another 2.
• 8 is even, divide by 2 once more: 8 ÷ 2 = 4. Record another 2.
• 4 is even, divide by 2 again: 4 ÷ 2 = 2. Record another 2.
• Finally, 2 divided by 2 equals 1. Record the last 2.

Bringing it all together, 32 can be expressed as the product of its prime factors: \[32 = 2 \times 2 \times 2 \times 2 \times 2 = 2^5\]In algebra, understanding prime factorization is useful because it allows us to simplify expressions, especially when dealing with radicals or exponents. When simplifying radicals, like in the exercise given, the prime factorization helps us break down the radicand into its simplest components. Then, we can apply the square root to each prime factor separately.

A square root is the opposite operation of squaring a number. Squaring a number means to multiply it by itself, and taking the square root asks the question: what number, when multiplied by itself, gives this number? For example, \(\sqrt{9} = 3\), because \(3 \times 3 = 9\).When dealing with square roots in algebra, especially with integers raised to a power, it is important to understand how they interact:

• \(\sqrt{a^2} = a\)
• \(\sqrt{a^3} = a \cdot \sqrt{a}\)

In the exercise, the square root of each prime factorization component is taken separately: \(\sqrt{(2^5)(t^5)(u^7)}\). This becomes:

• \(\sqrt{2^5} = 2^{5/2}\)
• \(\sqrt{t^5} = t^{5/2}\)
• \(\sqrt{u^7} = u^{7/2}\)

By examining these roots individually, you can then multiply the results together to achieve the simplified expression. This method of separating and conquering helps in maintaining clarity and accuracy while solving.

Exponents are compact ways to express repeated multiplication. When you see a number or variable with an exponent, it tells you how many times to multiply that number or variable by itself. For example, \(3^4\) means \(3 \times 3 \times 3 \times 3\).In algebra, exponents follow specific rules and properties:

• \(a^m \cdot a^n = a^{m+n}\)
• \((a^m)^n = a^{m\cdot n}\)
• \(a^0 = 1\)

To simplify expressions with both radicals and exponents, like \(\sqrt{32 t^{5} u^{7}}\), these powers are adjusted by the operation applied. The square root operation, denoted by \(\sqrt{}\), is equivalent to raising a number to the \(1/2\) power. From the exercise, each component was expressed using exponents:

• \(2^{5/2}\)
• \(t^{5/2}\)
• \(u^{7/2}\)

Understanding how to work with exponents is key to simplifying complex algebraic expressions. Raising to a power and finding roots are inverse relationships; mastering these concepts helps solve various mathematical problems efficiently.