Question

Simplify. $$ b^{2} \cdot b^{5} $$

Short answer

The simplified form of the expression \( b^{2} \cdot b^{5} \) is \( b^7 \).

Step-by-step solution

Identify the base and the exponents

In the expression \( b^{2} \cdot b^{5} \), the base is \(b\) and the exponents are 2 and 5 respectively.

Apply the law of exponents for multiplication

According to the law of exponents for multiplication, when you multiply powers with the same base, you add the exponents. Hence, \( b^{2} \cdot b^{5} = b^{2+5}\).

Simplify the expression

Adding the exponents 2 and 5 gives 7. Therefore, \( b^{2+5} = b^7\).

Key concepts

Simplifying Expressions

Simplifying algebraic expressions is about making them as straightforward as possible, usually to help understand the problem better or solve it more easily. In the case of our exercise with the expression \( b^{2} \cdot b^{5} \), simplification involves using the laws of exponents to combine the terms.

When we simplify expressions, especially those involving exponents, we aim to reduce them to their most basic form. This process often includes combining like terms, which are terms that have the same variables raised to the same power. By following specific rules, such as the law of exponents for multiplication, we ensure accuracy while making the expression more compact.

Exponent Multiplication

The law of exponent multiplication is essential for students to master because it applies to various mathematical operations where variables are raised to an exponent. According to this law, when you multiply two powers that have the same base, you can simply add the exponents together. This is portrayed in our example: \( b^{2} \cdot b^{5} \) becomes \( b^{2+5} = b^7 \).

It's crucial to understand that this rule only works when the base remains the same. The ability to combine exponents through addition makes complex expressions much more manageable, and is a foundational skill for algebraic manipulation and solving equations involving variables raised to powers.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables (like \(b\) in our example), and arithmetic operations. As part of learning to work with algebraic expressions, students should familiarize themselves with variable bases and the powers or exponents they're raised to.

In our exercise \( b^{2} \cdot b^{5} \), \(b\) is the variable base, and 2 and 5 are the exponents. Students learn to manipulate these expressions, such as by combining them through techniques like exponent multiplication, to solve for unknowns or simplify expressions for further calculations. Algebraic expressions are at the heart of algebra and are used to represent real-world situations mathematically, underlining their significance in education.