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Simplifying algebraic expressions is a fundamental skill in algebra which allows us to express equations in their most basic form. To do this, we combine like terms, use the laws of exponents, and perform any necessary operations.

For example, when we multiply expressions with exponents, like in the given exercise where we have the term \(x\) being multiplied by \(x^3\), we add their exponents because the bases are the same. The reason we can add the exponents is because of the law of exponents which states that \(x^a \times x^b = x^{a+b}\). So, \(x \times x^3 = x^{1+3} = x^4\).

Always remember, the key to simplification is to look for terms that can be combined, either through addition, subtraction, or, as in this case, by using the laws of exponents to simplify terms with the same base.

The process of multiplying numerators and denominators is straightforward when working with fractions. You treat the numerator and denominator separately, multiplying the top numbers together to get the new numerator and the bottom numbers together to get the new denominator.

In our example, the numerator is \(3x\) and \(2x^3\), and when we multiply them, we get \(6x^4\) due to the distributive property where \(3 \times 2 = 6\) and \((x \times x^3) = x^4\), as explained in the previous concept. Similarly, for denominators \(y\) and \(y^2\), we multiply them to get \(y^3\), since \(y \times y^2 = y^1 \times y^2 = y^{1+2} = y^3\).

It's essential to multiply carefully, ensuring each component of the numerator and denominator is accounted for, which will simplify the process of reducing the fraction in the next step.

Multiplying fractions is a basic arithmetic operation where two ratios are combined to form a single ratio. The concept is simple: Multiply the top numbers (numerators) together and multiply the bottom numbers (denominators) together.

The exercise given is a prime example of fraction multiplication with algebraic terms. The product formula for fractions is \(\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}\), regardless of whether the fractions involve numbers, variables, or a combination of both.

Applying the Rule

When we apply this rule to algebraic fractions, like in our exercise, \(\frac{3x}{y} \times \frac{2x^3}{y^2}\), we follow the same process:

• Multiply the numerators: \(3x \times 2x^3 = 6x^4\).
• Multiply the denominators: \(y \times y^2 = y^3\).
• Combine them to form a new fraction: \(\frac{6x^4}{y^3}\).

Remember, when multiplying fractions, you do not need to find a common denominator like you would when adding or subtracting fractions. The direct multiplication of numerators and denominators is all that's required.