Answer :
- Identify the common base: $x$.
- Apply the exponent rule for multiplication: $x^m
\cdot x^n = x^{m+n}$.
- Add the exponents: $3 + 4 = 7$.
- Simplify the expression: $\boxed{x^7}$.
### Explanation
1. Understanding the problem
We are asked to simplify the expression $x^3 \cdot x^4$. This involves using the rules of exponents. Specifically, when multiplying two terms with the same base, we add their exponents.
2. Applying the exponent rule
The rule for multiplying exponents with the same base is: $$a^m \cdot a^n = a^{m+n}$$ In our case, the base is $x$, and the exponents are $3$ and $4$.
3. Simplifying the expression
Adding the exponents, we have:$$3 + 4 = 7$$Therefore, the simplified expression is:$$x^3 \cdot x^4 = x^{3+4} = x^7$$
4. Final Answer
The simplified form of $x^3 \cdot x^4$ is $x^7$.
### Examples
Exponent rules are fundamental in many areas of mathematics and science. For example, in physics, when calculating the volume of a cube with side length $x$, if the side length doubles, the volume becomes $(2x)^3 = 8x^3$, which shows how the exponent affects the final result. Similarly, in computer science, exponents are used to describe the growth rate of algorithms, such as $O(n^2)$ or $O(2^n)$, where $n$ is the input size. Understanding exponent rules helps in analyzing and comparing the efficiency of different algorithms.
- Apply the exponent rule for multiplication: $x^m
\cdot x^n = x^{m+n}$.
- Add the exponents: $3 + 4 = 7$.
- Simplify the expression: $\boxed{x^7}$.
### Explanation
1. Understanding the problem
We are asked to simplify the expression $x^3 \cdot x^4$. This involves using the rules of exponents. Specifically, when multiplying two terms with the same base, we add their exponents.
2. Applying the exponent rule
The rule for multiplying exponents with the same base is: $$a^m \cdot a^n = a^{m+n}$$ In our case, the base is $x$, and the exponents are $3$ and $4$.
3. Simplifying the expression
Adding the exponents, we have:$$3 + 4 = 7$$Therefore, the simplified expression is:$$x^3 \cdot x^4 = x^{3+4} = x^7$$
4. Final Answer
The simplified form of $x^3 \cdot x^4$ is $x^7$.
### Examples
Exponent rules are fundamental in many areas of mathematics and science. For example, in physics, when calculating the volume of a cube with side length $x$, if the side length doubles, the volume becomes $(2x)^3 = 8x^3$, which shows how the exponent affects the final result. Similarly, in computer science, exponents are used to describe the growth rate of algorithms, such as $O(n^2)$ or $O(2^n)$, where $n$ is the input size. Understanding exponent rules helps in analyzing and comparing the efficiency of different algorithms.
