Question

Combine radicals, if possible. $$ 5 \sqrt{12 t^{3}}+2 t \sqrt{128 t}-3 t \sqrt{48 t} $$

Short answer

Question: Simplify and combine the like radicals in the following expression: $$ 5\sqrt{12t^{3}} + 2t\sqrt{128t} - 3t\sqrt{48t} $$ Answer: The simplified and combined expression is: $$ 10t\sqrt{3t} + 16t^2\sqrt{2t} - 12t^2\sqrt{3t} $$

Step-by-step solution

Simplify the radicals

To simplify the radicals, we need to break down the numbers and variables into their prime factors and then simplify. So, we have the following: $$ 5\sqrt{12t^{3}} = 5\sqrt{2^2\cdot3\cdot t^3} $$ The square root of \(2^2\) is 2, and the square root of \(t^3\) is \(t\sqrt{t}\). Thus, the first term simplifies to: $$ 5\cdot 2\cdot t\sqrt{3\cdot t} = 10t\sqrt{3t} $$ Now, let's simplify the other two terms: $$ 2t\sqrt{128t} = 2t\sqrt{2^7\cdot t} = 2t\cdot2^3\cdot t\sqrt{2t} = 16t^2\sqrt{2t} $$ $$ 3t\sqrt{48t} = 3t\sqrt{2^4\cdot3\cdot t} = 3t\cdot2^2\cdot t\sqrt{3t} = 12t^2\sqrt{3t} $$

Identify the like terms

Now, let's look at the simplified expressions: $$ 10t\sqrt{3t}, \quad 16t^2\sqrt{2t}, \quad -12t^2\sqrt{3t} $$ By comparing the terms, we can see that the first and the third terms are like terms as they both have the \(\sqrt{3t}\) factor. The second term does not have any like term in this expression.

Combine the like terms

Now, combine the like terms to get the final simplified expression: $$ (10t\sqrt{3t}) + 16t^2\sqrt{2t} + (-12t^2\sqrt{3t}) = 10t\sqrt{3t} + 16t^2\sqrt{2t} - 12t^2\sqrt{3t} $$ So, the final simplified expression for the given problem is: $$ 10t\sqrt{3t} + 16t^2\sqrt{2t} - 12t^2\sqrt{3t} \quad\_\square $$

Key concepts

Simplifying Radicals

Simplifying radicals involves reducing a radical expression to its simplest form. This can help you work with them more easily. Radicals are expressions that contain roots, such as square roots. To simplify, you must break down the number inside the radical to its prime factors and also reduce any variables within the radical.

• A simplified radical has no perfect square factors (other than 1) inside the radical sign.
• For example, the number 12 can be broken down into its prime factors: \(2^2 \cdot 3\).
• If any factor is a perfect square, like \(2^2\), it can be moved outside the radical as its square root (which is 2 in this case).

Variables can also be simplified if the exponent is greater than or equal to the index of the radical. E.g., \(t^3\) can be rewritten as \(t^2 \cdot t\), allowing some factors to be moved out of the square root as \(t \sqrt{t}\). This results in smaller numbers or simpler expressions outside the radical, making calculations easier.

Prime Factorization

Prime factorization involves breaking down a number into its prime number components. This method is fundamental in simplifying radicals. Prime numbers are numbers greater than 1 with only two positive divisors: 1 and the number itself. It helps to break complex expressions into manageable parts.

• Consider the number 48; it breaks down into \(2^4 \cdot 3\).
• Breaking expressions as such ensures you don’t miss any factor that can be easily dealt with outside the radical.
• It ends up simplifying radicals greatly by ensuring every term is expressed as a product of prime numbers.

To fully simplify expressions, prime factors reveal whether a number can be reduced under the radical sign, significantly easing the simplification process.

Like Terms Identification

In algebraic expressions involving radicals, identifying like terms is crucial for simplification. Like terms in radical expressions share the same radicand (the number inside the radical) and the same variables, with identical exponents outside of the radical.

• In \(10t\sqrt{3t}\) and \(-12t^2\sqrt{3t}\), both terms have \(\sqrt{3t}\) as their radicand, hence they are like terms.
• Unlike terms will have different radicands or variable components. For instance, \(16t^2\sqrt{2t}\) cannot be combined with another expression unless it also has a \(\sqrt{2t}\) factor with the same variable part.

Combining like terms means adding or subtracting their coefficients, allowing us to simplify the equation further. This simplification results in a cleaner and more concise expression.

Radical Expressions

Radical expressions are mathematical expressions that include a root symbol. These can come in the form of square roots, cube roots, and so forth. Handling them involves a few different operations as seen in the problem, including simplification and combination of terms.

• Simplifying a radical expression makes it easier to manage by breaking it down to its simplest form.
• Combining them involves the need to correctly simplify and identify common expressions that can be added or subtracted.

These expressions are crucial in intermediate mathematics and appear in various applications, like geometry and trigonometry, as well as more advanced mathematics. Mastering the techniques of simplifying, identifying like terms, and combining radical expressions can greatly benefit mathematical comprehension and performance.