Question

Multiply and simplify. $$\left(x^{2}\right)\left(x^{4}\right)$$

Short answer

\( x^{6} \)

Step-by-step solution

Understand the Law of Exponents for Multiplication

When multiplying two exponents with the same base, you can add the exponents together. This is expressed by the law: \( a^m \cdot a^n = a^{m+n} \). In this case, both bases are \( x \), and you will add their exponents.

Apply the Law of Exponents

Here you have \( x^{2} \) and \( x^{4} \). Using the law of exponents, you add the exponents because the bases are the same: \( 2 + 4 \).

Simplify the Expression

Now add the exponents to get the simplified expression: \( x^{2+4} = x^{6} \).

Key concepts

Exponent Multiplication

Exponent multiplication is an essential mathematical operation that allows us to simplify expressions involving exponents with the same base. The fundamental rule behind this is quite intuitive: when you multiply exponents with the same base, you keep the base and add the exponents. For example, multiplying x^2 by x^4 is not seen as a complex operation once you understand this principle. Instead of performing the multiplication separately, you can combine the exponents, resulting in x^{2+4} which simplifies to x^6. This operation becomes particularly useful in algebra and higher levels of mathematics, as it allows for quick simplification of expressions without lengthy calculations.

An additional tip for students is to visualize exponent multiplication: think of it as stacking layers. If x^2 represents two layers of x, and x^4 represents four layers, then putting these together, you have six total layers, hence x^6. Remember, this only works when the bases are the same; different bases require different methods.

Simplifying Expressions

Simplifying expressions is a fundamental skill in algebra that involves rewriting an expression in a simpler or more efficient form without changing its value. This is crucial as it makes equations easier to understand and solve. When dealing with exponents, simplification typically involves applying mathematical laws like the law of exponent multiplication. By knowing these rules, you can transform a potentially complicated expression into a more manageable one. For instance, rather than multiplying x^2 by x^4 for each value of x, simplifying the expression to x^6 saves time and reduces the possibility of error.

For students aiming to master simplifying expressions, it is important to practice regularly and become familiar with common patterns. Also, always look for opportunities to combine like terms and apply exponent laws, as these are the shortcuts that make algebra more efficient and manageable. Simplifying is not just about making an expression shorter; it's about making it clearer and more directly solvable.

Mathematical Laws

Mathematical laws are the frameworks that govern the operations and relationships within mathematics. They provide the structure needed for working with numbers and expressions consistently and accurately. One such law is the Law of Exponents, specifically applied when you're dealing with exponent multiplication. By following this law, x^2 times x^4 becomes x^{2+4} and ultimately x^6 because it is understood that when bases are the same, we must add the exponents. This is only one example out of many laws which include the distributive property, commutative and associative laws among others.

Understanding and applying these mathematical laws are vital for anyone learning math. Tips for students include creating summaries or flashcards of these laws and practicing problems that apply them in various contexts. Having a solid grasp of these laws not only aids in simplification but also in the comprehension of more advanced mathematical concepts. It allows students to approach problems methodically, based on a set of rules that are universally recognized in the realm of mathematics.