Problem 27 - Contemporary Precalculus

Question

Write the given expression without using radicals. $$\sqrt[5]{t} \sqrt{16 t^{5}}$$

Step-by-Step Solution

Answer

Question: Simplify the expression $$\sqrt[5]{t}\sqrt{16 t^{5}}$$ without using radicals. Answer: $$4t^{\frac{26}{10}}$$

Step 1: Rewrite the roots using exponents

In order to deal with the roots, we'll first rewrite them using fractional exponents. The nth root of a number x can be expressed as x^(1/n). Rewrite the given expression using exponents: $$\sqrt[5]{t}\sqrt{16 t^{5}} = t^{\frac{1}{5}}(16t^5)^{\frac{1}{2}}$$

Step 2: Simplify the square root expression

Now, we'll focus on simplifying the square root part of the expression \((16t^5)^{\frac{1}{2}}\). Using the rules of exponents, we distribute the exponent to each term inside parentheses: $$\left(16t^5\right)^{\frac{1}{2}} = 16^{\frac{1}{2}}\left(t^5\right)^{\frac{1}{2}}$$

Step 3: Simplify the exponents

We'll now simplify the exponents by using the rule (a^m)^n = a^(mn): $$16^{\frac{1}{2}}\left(t^5\right)^{\frac{1}{2}} = 2^{\frac{4}{2}} \cdot t^{\frac{5}{2}}$$ $$ = 2^2 \cdot t^{\frac{5}{2}}$$ $$ = 4t^{\frac{5}{2}}$$

Step 4: Combine the two expressions

Now, we'll multiply the two simplified expressions \(t^{\frac{1}{5}}\) and \(4t^{\frac{5}{2}}\): $$t^{\frac{1}{5}} \cdot 4t^{\frac{5}{2}}$$

Step 5: Simplify by adding exponents

Since we are multiplying terms with the same base, namely 't', we can add the exponents according to the rule a^m*a^n=a^(m+n): $$t^{\frac{1}{5}} \cdot 4t^{\frac{5}{2}} = 4t^{\frac{1}{5}+\frac{5}{2}}$$ $$= 4t^{\frac{1+25}{10}} = 4t^{\frac{26}{10}}$$

Step 6: Final answer

The given expression without radicals is: $$4t^{\frac{26}{10}}$$

Key Concepts

Simplifying Expressions

Simplifying expressions involves rewriting complex mathematical terms in a more manageable form, without changing their value. For example, we often rewrite expressions to eliminate radicals or simplify fractions. This makes calculations easier and results more readable.

• Identify complexity: Before you simplify, identify parts of the expression that can be rewritten or reduced. Look for roots, powers, or fractions.
• Use consistent notation: Ensure consistency by converting all terms to a similar form, such as using fractional exponents instead of radicals.
• Apply algebraic rules: Use rules of arithmetic and algebra, like the distributive property or combining like terms, to simplify the expression further.

Breaking an expression into smaller, simpler parts helps understand the overall problem and makes solving it more straightforward.

Rules of Exponents

Exponents have specific rules that make calculations simpler. They help to manipulate terms raised to a power efficiently. Here are key rules to remember:

• Product of powers rule: When multiplying like bases, add their exponents. For example, \(a^m \cdot a^n = a^{m+n}\).
• Quotient of powers rule: When dividing like bases, subtract the exponents. Written as \(a^m / a^n = a^{m-n}\).
• Power of a power rule: When raising a power to another power, multiply the exponents, \((a^m)^n = a^{mn}\).
• Zero exponent rule: Any non-zero number raised to the power of zero equals one, \(a^0 = 1\).

Remembering these rules is crucial for simplifying expressions involving exponents.

Fractional Exponent Notation

Fractional exponent notation is another way to express roots, providing a link between roots and powers. It allows for easier manipulation and simplification of expressions. Here's how it works:

• Fractional exponents express roots: \(x^{1/n}\) is equivalent to \(\sqrt[n]{x}\). For example, \(x^{1/2}\) represents the square root of x.
• Combining powers and roots: Multiply the fractional exponent by a whole number to combine powers with roots, such as \(x^{m/n}\), which is \((x^m)^{1/n}\) or \(\sqrt[n]{x^m}\).
• Facilitating simplification: Using fractional exponents allows easy application of exponent rules, letting us simplify complex root expressions systematically.

This method bridges the gap between radical and exponential expressions, simplifying computations in algebra.

Roots and Radicals

Roots and radicals represent numbers raised to fractional powers, such as square roots or cube roots. Understanding them is key to many algebraic simplifications.

• Radicals as roots: The notation \(\sqrt[n]{x}\) means the nth root of x. The most common radical is the square root, represented as \(\sqrt{x}\).
• Link to fractional exponents: The nth root of x can be rewritten as \(x^{1/n}\). This connection allows for easier manipulation when combined with powers.
• Simplification using properties: Combine and simplify radicals by using properties of exponents, such as \((x^m)^{1/n} = x^{m/n}\).

Being comfortable with converting between radicals and fractional exponents can make complex algebraic expressions more accessible. This understanding facilitates easier computation, transforming cumbersome radicals into manageable exponent forms.